Based on his words and actions in "After Twenty Years," how would you describe Bob's personality?

Bob can be described as determined. Twenty years ago, he left New York for the West determined to be successful. His friend, Jimmy, chose to stay in his familiar surroundings. When asked by the cop if he'll continue to wait past ten o'clock, Bob says he will. He is determined to see his old friend.

Confident is another word to describe Bob's personality. Bob immediately informs the cop of the reason for his presence. Although he has not earned his success by entirely legal means, Bob doesn't hesitate in speaking to the cop and telling his story. Bob is also confident in his belief that Jimmy will arrive at their meeting spot if he "is alive on earth."

Finally, Bob is loyal. He remembers his friendship with Jimmy fondly and hopes to see his old friend again. Jimmy repays the loyalty to the extent that he can. He sends another officer to arrest Bob instead of arresting his friend himself.

When the policeman first accosts Bob he "spoke up quickly" without being asked and tells the officer that he is meeting an old friend, adding,

"I'll explain if you'd like to make certain it's all straight."

This hasty voluntary disclosure indicates that Bob wants to appear forthright and honest. Further, he informs the policeman that he went out West to make his fortune, but he adds that he and his old friend "lost track of each other" as he was "hustling around."  Then, he pulls out a beautiful watch that is covered with diamonds, suggesting that, perhaps, he has been involved in something illegal.

However, all the time that 'Silky Bob' converses with the policeman, he praises his friend Jimmy and states that he will wait at least thirty extra minutes for his friend. Clearly, there is an affection for Jimmy that Bob yet possesses, an affection that Jimmy, too, retains because he cannot bring himself to arrest 'Silky Bob.'

What does the sentence suggest about Bob's personality in "After Twenty Years"?

The quote you mentioned—especially the words "his egotism enlarged by success"—tells us much about Bob's personality and character. Those words highlight Bob's materialistic predilections. Essentially, Bob's sense of self is predicated upon his ability to gain material success. In that light, he's extremely proud of the fact that he excelled in the competitive business world of the West. His pride in his accomplishments is clear, and he can't wait to tell his old friend Jimmy Wells all about it.

When he meets the man he thinks is Jimmy, Bob excitedly "outlines the history of his career." He thinks that he has bested Jimmy in life, and he can hardly hold himself back from openly boasting about his success. Bob is so preoccupied with his telling that he almost fails to realize that the man he is speaking to isn't Jimmy.

When he finally snaps "You're not Jimmy Wells," it's too late. The plainclothes police officer tells Bob that he has been under arrest for ten minutes. So, the quote basically tells us this: Bob's primary purpose in life is to gain material success. His ego is predicated on his ability to outperform his business competitors, and he enjoys boasting about his exploits after the fact.

Who is the main character in "After Twenty Years" and how would you characterize Bob?

The two main characters in the story are the two men who used to be such good friends -- Jimmy Wells and Bob (whose last name we do not know).  I think you could argue for each of these men being the major character.  I will say that I think Jimmy is truly the major character because the story turns on what he actually does in the story(whereas Bob is only important for what he has already done before the story).

Bob has become a criminal so in that way he is bad.  On the other hand, he remains good enough of a person to remember Jimmy fondly and want to keep their meeting.

Who is the character Bob in "After Twenty Years" by O. Henry?

O. Henry's touching short story with the mildly surprising ending is about two old friends, twenty years, and life's vagaries and changes. Bob and Jimmy Wells at the ages of eighteen and twenty were set to go off on their separate ways to make their fortunes and find their destinies. They agree to meet back together in twenty years from the date and time of their parting to see each other again. A survey of the movement of the story will help sort out what Bob is.

Both are New York boys, Bob has an ambition to meet his destiny and make his fortune in the West while Jimmy desires to remain firmly rooted in New York City. As Bob tells the city policeman patrolling the street where Bob and Jimmy are to meet, Bob moved around the West quite a bit and the two friends eventually quit corresponding over the years:

the West is a pretty big proposition, and I kept hustling around over it pretty lively.

Bob also says he knows that if Jimmy is still alive, he will show up at the appointed place and confirms that he will wait if Jimmy is late in coming. When a tall man huddled in an overcoat approaches, he and Bob are reunited and start to walk "around to a place" for a meal.

In the light of a drug store window, Bob sees that the man isn't really Jimmy. Here is where we discover--if we weren't suspicious of it already because of all the mentions of the West and all the diamonds sparkling on Bob and his possessions--that Bob is a criminal from Chicago who is wanted by the Chicago police. Jimmy turns out to be an impostor come to arrest him.

The association of great wealth, criminal activity, historical time period (c. 1904), and the city of Chicago lead the reader (especially in O. Henry's day because it was a contemporaneous time period) to understand that Bob is part of the Chicago gangster scene, reminiscent of the slightly later Al Capone. Jimmy Wells, of course, was the patrol officer with whom Bob spoke, as his note explains:

"Bob: I was at the appointed place on time. When you struck the match to light your cigar I saw it was the face of the man wanted in Chicago. Somehow I couldn't do it [arrest you] myself, so I went around and got a plain clothes [police] man to do the job. JIMMY.”

The author does not seem either to like or dislike his character Bob. His description of Bob and quotation of his dialogue are objective and neutral, although the reader gets the distinct impression that Bob is a materialistic and unscrupulous opportunist. The reader does not get any impression of how the police officer may be sizing the other man up. The policeman seems objective, like the author. The reader, of course, does not realize that the uniformed beat cop is Jimmy Wells, the man Bob is waiting to meet after twenty years. If O. Henry had adopted a more negative tone in depicting Bob, the reader might infer that this antipathy was a reflection of Jimmy Wells' personal reaction to his old friend. And that might give away the big surprise in the story, which is that Bob is talking to his old friend Jimmy Wells without knowing it.  

Bob is flashy and superficial, but he does not say or do anything to make him seem unlikeable. The reader can actually identify with him because he is being confronted by a cop, "a fine picture of a guardian of the peace"--and most readers have been in that situation at one time or another, even if they have only been stopped for speeding. O. Henry planned to have Bob get busted, but he didn't want his reader to feel glad that this was happening to the fugitive. The reader is more likely to share Bob's feelings of humiliation at letting himself get caught after volunteering so much information about himself and his activities, along with with Bob's feeling of bitter disappointment and chagrin. Bob is depicted as being full of pride and self-confidence at the beginning of the tale and totally deflated and demoralized at the end, in spite of his diamond scarf-pin and his diamond-studded gold watch. The reader can only imagine that Jimmy must feel some remorse about betraying an old friend but satisfaction in having performed his duty "by the book."