Zaroff hunts him. During the hunt, Rainsford reminds himself over and over again, "I must keep my nerve," as a way to stay off fear of losing his life. When Rainsford realizes Zaroff allows him to live another day simply for the sport of it, the text says, "Then it was that Rainsford knew the full meaning ofterror." The word "terror" suggests a more heightened sense of fear for one's life. Another time Rainsford feels afraid is after he attacks Zaroff with the Malay man-catcher. Zaroff's response causes Rainsford anxiety as follows:

He stood there, rubbing his injured shoulder, and Rainsford, with fear again gripping his heart, heard the general's mocking laugh ring through the jungle.

It isn't until after the third day that Zaroff may feel fear, although he is really good at not showing it. When Zaroff finds Rainsford in his room, he seems more startled than fearful when he screams "Rainsford... How in God's name did you get here?"

In conclusion, the only fear these men might be able to describe is the fear of losing their lives. They are men who are not used to feeling fear because they would be the ones to incite fear in others. General Zaroff uses the word "thrill" when discussing how he feels when faced with a wild animal, so his definition would include something where he feels an adrenaline rush. Rainsford, on the other hand, fears for his life as he is hunted. Another word used to define how he feels during the hunt is "terror," so his definition would include anything that makes him feel terrified for his life. In the end, both experience fear for their lives, but it seems these professional hunters only understand fear when their lives are truly threatened.

The only thing that General Zaroff is afraid of is boredom.

General Zaroff admits that he is a bit savage.  After all, he is a Cossak.  Zaroff has decided to make his home on Ship-trap Island and stock it with sailors and imported men he can use as hunting prey.

"Oh," said the general, "it supplies me with the most exciting hunting in the world. No other hunting compares with it for an instant. Every day I hunt, and I never grow bored now, for I have a quarry with which I can match my wits."

The problem is that ordinary animals cannot keep Zaroff’s interest anymore.  He is too skilled a hunter to be challenged by a jaguar or elephant.  He apparently hates being bored.  He calls it his “complaint” and refers to it as “ennui” and boredom.  He craves excitement like a drug.  He thinks Rainsford is his ticket to exhilaration.  In the end, Rainsford does cause him a challenge, because Rainsford wins.  When he finds Rasinford in his room, he is excited and describes it as “splendid” that they will fight—but Rainsdford wins.

General Zaroff is someone whose entire life is defined by his excellence as a hunter. As he himself tells Rainsford,

God makes some men poets. Some He makes kings, some beggars. Me He made a hunter.

But in such a context, what does it mean when someone starts to lose interest in the very pursuit that gave their life meaning? Where does one go from there?

In a very real respect, Zaroff became too masterful a hunter for his own good. Hunting even the most dangerous of animals began to become too easy, to the point where it was becoming boring. This created a psychological crisis (given that hunting was ultimately what sustained and defined his entire existence). It was out of that crisis, looking to find a more challenging quarry, that he began hunting humans instead.

However, that earlier crisis has not been entirely resolved. He confesses as much to Rainsford, shortly before enlisting Rainsford himself to be his quarry. Admitting that his previous night's hunt had been a bitter disappointment, Zaroff tells Rainsford, "I detected traces of my old complaint." Thus, we can infer that what Zaroff fears most (and what still torments him) is this: hunting humans is only a temporary solution, and eventually he'll grow bored of that as well.

The quick answer to this is that Zaroff is worried about what he calls "ennui."  This if a French word that means something like boredom.  So, Zaroff is worried because he is getting bored with his life.

What is making him bored is the fact that his hunts have not been all that fun.  He complains to Rainsford that the last man he hunted was not very interesting.  The man was a sailor and he apparently just walked or ran in a straight line and was too easy to follow.

What Zaroff is saying is that he needs a challenge.  To me, this is implying that he is going to want to hunt Rainsford.  He will figure that Rainsford would make good prey because he is an intelligent man who has done a lot of hunting.

So, the quick answer is that Zaroff is worried about being bored and he is bored because his prey has not been elusive enough.