everything fly back to seed, flee death, rush down to their beginnings, suns rise in western skies and set in glorious easts, moons eat themselves opposite to the custom, all and everything cupping one in another like Chinese boxes

Eckels likes the idea that his money can buy him the right to shoot an animal from the past and initially has little understanding as Travis tries to explain to him the dangers of changing even the smallest event from long ago.

Eckels, used to being in charge, doesn't like to be told what to do. He flushes (a sign of his anger) back in the dinosaur age when Travis tells him to put down the gun he is aiming playfully. Eckels tries to regain control by asserting possession, asking Travis "Where's our Tyrannosaurus?”

When he sees it, he decides for himself that it can't be killed, showing he is used to consulting his own opinion and nobody else's. If he thinks something, it must be true. He's not used to being deferential or humble:

“It can’t be killed.” Eckels pronounced this verdict quietly, as if there could be no argument. He had weighed the evidence and this was his considered opinion.

Of course, he is wrong. He also overestimates himself: the Tyrannosaurus frightens him much more than he expected, and this causes him to leave the path and enter the jungle.

His reaction to the dinosaur and his mistake in killing a butterfly humbles him to the point he apologizes, but it is too late. His arrogance has already changed history.

In "A Sound of Thunder", Eckels is apparently a rich, leisured hunter who pursues rare experiences and novelties. This leads him to purchase an expensive time safari, in which he will be taken into the distant past via a time machine, to hunt animals that are extinct in the modern era. 

Two significant elements flesh out some of Eckels' personality; his reliance on money to get what he wants, and his lack of insight. 

His relationship with money seems indistinct at first - we merely know that he has paid a large amount of it to purchase the trip. We might assume that a person to whom money is more significant might have better researched what he was purchasing, but Eckels is full of questions and apparent ignorance, which may indicate that he simply pursued the novelty offered by the time safari company without a second thought. Later, his true relationship with money is revealed when he offers to buy his way into the good graces of the safari leaders after he has violated their rules and probably altered the timeline, not realizing that money is an irrelevant compensation for such an act.

Eckels is basically rich but ignorant, and like some of Bradbury's other stories such as "The Veldt" he serves as a warning about allowing the comfort, convenience and novelty of technology and civilized lifestyles to overwhelm and stagnate our self-control and sensibility.

Eckels is a pompous blowhard.  That might be not be technical enough, but it is how I would describe him.  Perhaps a better character description of Eckels would be to describe him as an arrogant coward.  

Right from the beginning of the story, Bradbury paints Eckels as a guy that the reader takes an aversion to.  

The muscles around  his mouth formed a smile as he put his hand slowly out upon the air, and in that hand waved a check for ten thousand dollars to the man behind the desk.

He comes walking into Time Safari Inc. like he owns the place.  He's waving around his big check like he is something special and better than the "lowly" workers that will be in charge of his life.  Eckels is a man who thinks that the world revolves around him.  Take this quote for example: 

"Shooting my dinosaur," Eckels finished it for him.

Eckels thinks that the entire safari into the past is for him and only him.  Keep in mind that there are several other hunters that go back in time with Eckels.  

Even once the time machine enters the past, Eckels still has not grasped the danger of the very real situation that he is in.  He brags about how many other safaris that he has been on and how many animals that he has killed.  He even plays with his gun like it is a mere toy to him.  

Eckels, balanced on the narrow Path, aimed his rifle playfully.

When it comes down to it though, Eckels is a coward.  Granted, a T. rex would be a scary sight, but Eckels is the only hunter who so completely loses his self control that he is unable to follow a simple path back to the time machine.  

"Get me out of here," said Eckels. "It was never like this before. I was always sure I'd come through alive. I had good guides, good safaris, and safety. This time, I figured wrong. I've met my  match and admit it. This is too much for me to get hold of." 

"Don't run," said Lesperance. "Turn around. Hide in the Machine." 

"Yes." Eckels seemed to be numb. He looked at his feet as if trying to make them move. He gave a grunt of helplessness. . . 

Travis shook his hand away. "This fool nearly killed us. But it isn't that so  much, no. It's his shoes! Look at them! He ran off the Path."

Eckels is shot by Travis because he stepped on the butterfly and changed the future.

At the beginning of the story, Eckels asks a prophetic question.

"Does this safari guarantee I come back alive?"

Of course, they tell him they can’t.  He is nervous about hunting dinosaurs, but they explain the process and he goes along.  He is warned not to step off the path, but he accidentally kills a butterfly.  Travis is very angry. 

When they return to the present, Eckles finds out that the spelling on the sign has changed and a different person has been elected president.  Since this man is supposed to be a tyrant, we get the idea that Eckels has messed up the future.

Eckels moans, wanting to know if he can go back to the past and fix his mistake.  Travis says nothing, but answers with his gun.

He did not move. Eyes shut, he waited, shivering. He heard Travis breathe loud in the room; he heard Travis shift his rifle, click the safety catch, and raise the weapon.

Eckels is shot more in revenge than punishment, because there is nothing that can be done.  Simply by killing a butterfly, he has altered the future significantly.  It is a lesson in how actions can have widespread unintended unforeseeable consequences.

I would not describe Eckels in very positive terms.  I have never liked the character of Eckels.  The fact that he messes everything up in the present doesn't exactly endear him to me.  

I would mainly describe Eckels as arrogant and selfish.  My feelings about arrogance start right from the beginning of the story.  

"The muscles around his mouth formed a smile as he put his hand slowly out upon the air, and in that hand waved a check for ten thousand dollars to the man behind the desk."

Apparently time travel isn't cheap in the year 2055.  No problem there.  But most people would approach the desk and present their check.  Not Eckels.  He thinks he has to wave his money around for all to see.  Like it's a banner of privilege and status or something.  

I get feelings of selfishness from three specific lines.  First, this one: 

"Does this safari guarantee I come back alive?"

Notice that Eckels wants a guarantee on his life.  I understand that, but I get the feeling that Eckels only cares about his life and not the lives of the other hunters and guides.  He could have asked "Does this safari guarantee we come back alive?"  The sentiment would have been the same, but Eckels wouldn't appear like a selfish rich kid. 

The second line of selfishness is this line:

All you got to worry about is-"

"Shooting my dinosaur," Eckels finished it for him.

Eckels is so focused on what's his that he even has to interrupt the Safari Inc. employee.  

The last support for Eckels being selfish (and a bit ignorant) is when the rules are being explained to him about the possible repercussions of time travel.  Eckels just doesn't get it and/or doesn't seem to care.  

"And all the families of the families of the families of that one mouse! With a stamp of your foot, you annihilate first one, then a dozen, then a thousand, a million, a billion possible mice!"

"So they're dead," said Eckels. "So what?"

Every time that I read Eckel's response, I hear him speaking the line in a confrontational tone.  I don't hear him asking an actual question of concern.  Eckels genuinely doesn't care if those mice are dead, because he is not a mouse.  It doesn't concern him, so he doesn't care about those lives.  If he doesn't care about those lives, why would he care about the lives of the other members of the time travelling group?  Eckels is an arrogant and selfish rich brat. 

Eckels is a nervous man.

Eckels is a big game hunter.  He hires Time Safari, Inc. to take him back to the days of the dinosaurs so he can shoot a Tyrannosaurus Rex.  However, Eckels is a bit of a nervous man.  He does not feel confident in his abilities, and gets worried even before he travels back in time.  This quote demonstrates that Eckels was nervous even before facing the dinosaur.

A warm phlegm gathered in Eckels’s throat; he swallowed and pushed it down. The muscles around his mouth formed a smile as he put his hand slowly out upon the air, and in that hand waved a check for ten thousand dollars to the man behind the desk.

Eckels asks them if they will guarantee that he comes back alive.  They tell him they guarantee nothing, and yet he still goes.  He is nervous, but doesn’t back out.

Eckels is fascinated with time travel.

One of the reasons Eckels agreed to go despite his nervousness might be that he seems completely fascinated with time travel itself.  He is willing to pay the thousand dollars just for a chance to get in a time machine.  This is why he doesn’t back out, even after they won’t guarantee his safety.

“Unbelievable.” Eckels breathed, the light of the Machine on his thin face. “A real Time Machine.” He shook his head. “Makes you think. If the election had gone badly yesterday, I might be here now running away from the results. Thank God Keith won. He’ll make a fine President of the United States.”

I guess to Eckels time travel is kind of like a roller coaster.  It is so scary that it’s exciting. He has come this far, and he can’t back down.  He wants to see what will happen.

Eckels is self-centered.

When Travis goes into great detail to explain to Eckels why he can’t change the past, he seems a little dense.  He wants a longer, drawn-out explanation.  He doesn’t really seem to care about anything but himself, and certainly not small animals.

“So they’re dead,” said Eckels. “So what?”

“So what?” Travis snorted quietly. “Well, what about the foxes that’ll need those mice to survive?... Fifty-nine million years later, a cave man, one of a dozen in the entire world, goes hunting wild boar or saber-toothed tiger for food. But you, friend, have stepped on all the tigers in that region. By stepping on one single mouse …”

If Eckels had paid more attention to this explanation, maybe he would have been more careful when he stepped off the path.  Even after he wanders off, Eckels doesn’t really seem to get it.  He complains that it is just a little butterfly he stepped on.  When they return to the present, he doesn’t really have an opportunity to ponder his mistake, because Travis shoots him.

Eckels is the hunter who panics and causes the permanent alteration of history in Ray Bradbury's science fiction short story "A Sound of Thunder." Going back in time to witness the hunt of the dreaded Tyrannosaurus Rex, Eckels strays from the designated path and, upon returning to the present time, realizes he has crushed a butterfly and it has stuck to his boots. The tiny butterfly's untimely death has caused a chain of events that includes visible changes in people, their language, and in the recent presidential election. After Eckels' discovery, the tour guide and hunter Travis adjusts his rifle and raises the weapon. Next is heard the sound of thunder, the original description of the sound of a charging T-Rex. But this sound of thunder is different: It is the sound of Travis's bullet killing Eckels, the fitting end for his unalterable mistake in the past.

A rather self-serving wealthy man, Eckels contracts with Time Safari, Inc. to take him in a time machine in order to have the thrill of shooting the Tyrannosaurus Rex, one of the dinosaurs of the beginnings of the world. This time travel involves returning to a site in which the creature killed died without disturbing the balance of nature.  For instance, the Tyrannosaurus that Eckels will shoot has died from a falling tree, so killing him will have no other effect that his original death, and, thus, not alter the course of time.

What is of utmost importance in this return to a past time is that nothing be disturbed other than the dinosaur to be killed. Therefore, Eckels is given strict instructions to remain upon the company's anti-gravity path so that no dirt, no grass, no insect, etc. is disturbed or picked up on his shoes or person.

A touch of the hand and this burning would, on the instant, beautifully reverse itself.  Eckels remembered the wording in the advertisements to the letter.

Because this time travel is so dangerous and exacting, Eckels must pay a great sum of money in order to travel back in time. If anything happens that is not according to regulations, he will also have to pay another ten thousand dollars.

Although Bradbury does not specifically state what happens to Eckles, we can assume that Eckles is killed by Travis.  Discovering that Eckles has indeed brought back something from the future and the course of history has been changed the end of the story has Eckles on his knees with his eyes closed and hears Travis move his rifle, take off the safety and then a sound of thunder, or as I understand it to mean, the firing of the gun.

Eckels is the man who changes the present. . . by changing the past . . . because as George Orwell wrote, "whoever controls the past, controls the present."  

The first thing to know about this story is that time travel into the past is possible.  In fact, it's common enough where it has become a tourism industry.  The story begins when Eckels walks into the offices of Time Safari Inc.  He wants to go back in time to hunt and kill a dinosaur.  As Eckels learns about the rules, so does the reader.  The time travel company will take him back in time to the time of dinosaurs.  The time travel company also provides a guide and a really cool floating sidewalk thing that doesn't touch anything in the past.  

“And that,” he said, “is the Path, laid by Time Safari for your use. It floats six inches above the earth. Doesn’t touch so much as one grass blade, flower, or tree. It’s an anti-gravity metal. Its purpose is to keep you from touching this world of the Past in any way. Stay on the Path. Don’t go off it. I repeat. Don’t go off. For any reason! If you fall off, there’s a penalty. And don’t shoot any animal we don’t okay.

The guide is to ensure that Eckels kills the correct dinosaur.  The dino in question is a dinosaur that is about to die anyway, so Eckels actions won't actually change anything in the past that wasn't already going to happen soon after.  

“Right,” said Lesperance. “I track them through their entire existence, noting which of them lives longest. Very few. How many times they mate. Not often. Life’s short. When I find one that’s going to die when a tree falls on him, or one that drowns in a tar pit, I note the exact hour, minute, and second. I shoot a paint bomb. It leaves a red patch on his side. We can’t miss it. Then I correlate our arrival in the Past so that we meet the Monster not more than two minutes before he would have died anyway. This way, we kill only animals with no future, that are never going to mate again. You see how careful we are?”

The time travel company doesn't want to cause a ripple effect through time by doing something in the past that would change thousands of generations of life.  

“say we accidentally kill one mouse here. That means all the future families of this one particular mouse are destroyed, right?”

Unfortunately for the story, Eckels gets scared of the dinosaur that he is supposed to shoot.  He accidentally steps off of the path and kills a butterfly.  When everybody returns to the present, the entire United States is a militarily controlled dictatorship. 

Eckels, not looking back, walked blindly to the edge of the Path, his gun limp in his arms, stepped off the Path, and walked, not knowing it, in the jungle. His feet sank into green moss. . . Eckels felt himself fall into a chair. He fumbled crazily at the thick slime on his boots. He held up a clod of dirt, trembling, “No, it can’t be. Not a little thing like that. No!”

Embedded in the mud, glistening green and gold and black, was a butterfly, very beautiful and very dead."

It's Eckels who helped create the now familiar phrase "The Butterfly Effect."  The phrase is referencing how one small change can have huge consequences.  

The story is generally about Eckels' trip on a "time safari" - going into the distant past via a time machine to hunt extinct animals. However, when the hunting party encounters the tyrannosaur that Eckels is intended to shoot, he loses his nerve and panics, falling off of the designated Path and accidentally crushing a butterfly. 

Via an earlier series of expositions, it is explained that virtually any alteration to the past has the potential to cause catastrophic changes to the timeline, altering it in ways that no one can anticipate because they are so complex and difficult to track. Thus Eckels' error is immediately met with threats to kill him or leave him behind; he has not only endangered the company, but the course of human history.

When the group returns to the current time, it is clear that Eckels did indeed alter the flow of history; English is now written differently, and a politician that all of the characters disliked has been elected president. It is implied that this timeline is a stupid, simple one compared to the one they came from. It is then strongly suggested that Travis shoots Eckels as a form of justice or revenge for what he has done.