You are asking about one of my favorite topics ever. Time travel and paradox consequences. I'm a real nerd for that topic. The "butterfly effect" is simply this. Small changes in the past can create massive changes in the future. Bradbury is not the first person to suggest that kind of past/future correlation, and interestingly enough, Bradbury is not credited with making the phrase "butterfly effect" famous either. That credit goes to Edward Lorenz. Lorenz was a meteorologist during the mid 1900's. During one of his weather model simulations, Lorenz changed a minuscule amount of data in the prediction model. The result was a completely different outcome than the original model. Lorenz then made the imaginative connection that a single flap of a butterfly wing could change an entire weather pattern -- thus, the butterfly effect. A small change in one location, creates drastic differences elsewhere.
In Bradbury's story, "A Sound of Thunder," Eckels steps off of the path while hunting dinosaurs millions of years ago. Unfortunately, his foot stepped on and killed a butterfly. That butterfly's entire line of descendants were instantly wiped out, which cancelled out any interactions they might have had with anything else in the world. The result was that the language in Eckels own time was changed and the system of government was altered as well.
His face was cold. His mouth trembled, asking: “Who - who won the presidential election yesterday?”
The man behind the desk laughed. “You joking? You know very well. Deutscher, of course! Who else? Not that fool weakling Keith. We got an iron man now, a man with guts!”
For my own example, I like to use the original Back to the Future movie. In that movie, Marty McFly changes one small event with his father. His mom then falls in love with Marty instead of his father, and Marty starts slowly disappearing from the world.
A different example is my favorite paradox. It's called the grandfather paradox. Suppose that you went back in time and killed your grandfather before he had a chance to have your father. That means with no father of your own, you don't exist. Right? Well, if that is the case, then who went back in time to kill your grandfather?