What kind of humor did people generally enjoy in the Roman Empire (say, around 0 BC). How did it differ from our humor in the modern world?

There is no year 0, that's why the new millenium started in 2001 not 2000.

Anyway, moving on to more important things, it's necessary to divide your question into two different parts. First of all, the public display of humor, such as in the theater and so forth, and the more private form of humor. There's not really any difference between them except for our sources of evidence, and there isn't any real reason based on what we've got to suggest that the classes had different forms of humor. The only difference was the directness of the address, and the method used to spread it--there wasn't really any whitewashing of profanity or lewdness in public address (hell, even Cicero makes really nasty and lewd jokes about Clodius Pulcher and his sister).

Our best evidence for humor of any kind would of course have to be within the comic plays and their more popular (at least under the Principate) counterparts, pantomimes and mimes. The Romans had pretty much no knowledge of Athenian Old Comedy (the stuff of Aristophanes) outside of any books that wealthy collectors had, and we have no record that they ever staged any of the Old Comedies. The Romans, it seemed, didn't particularly like the style of humor in Old Comedy, which was bawdy, grotesque, lewd, and quite bold in its willingness to mercilessly lampoon important public figures (the only example of restraint in the latter that we have is the fact that Aristophanes finds it necessary in his early plays to use pseudonyms for Cleon, a practice that he abandoned later, when he was so thoroughly disgusted with Cleon that he had lost all respect for him). Instead, the Romans staged mainly translated adaptations of New Comedy, the stuff of Menander (who was the favorite). The earliest form of these, the fabula palliata, replaced an earlier form of Roman comedy which had developed out of Livius Andronicus and Gnaeus Naevius under the influence of Greek theater, but largely without Greek subject matter (and possibly also in traditional Roman meters). These early plays were replaced in less than 50 years by the fabula palliata, brought by Terence and Plautus, which soon became not only the dominant form of comic theater, but the dominant theater in general (unlike the Athenian, who preferred tragedy). Like their New Comedy models, the fabula palliata were known for intricate plots, with a set of stock characters making mostly oblique jokes about social practices. New Comedy broke sharply from Old Comedy in its refusal to insert political, literary, and social criticism into the work, preferring to make rather light and witty jokes about silly social practices (this can be seen in New Comedy's modern descendent, the comedies of Shakespeare, who used Terence and Plautus as his models and occassionally outright adapted their work) Although it could be downright lewd at times (one of the favorite characters in New Comedy was the lewd courtesan, who often would perform some sort of burlesque) New Comedy was a great deal more polite than the Old Comedies, which in addition to constantly insulting people was not above fart jokes, shit jokes, sex jokes (all the time with these three, all the time), and an awful lot of penis jokes (reasonably, since the comic theater was originally tied to certain fertility rituals and the actors still wore big leather phalli). A few characters in New Comedy (the fan favorites, mostly) would have somewhat bawdy lines (mostly about hookers), but for the most part it depended on wit and social satire.

The other main form of comic theater was the mime. This is not the kind of thing we associate with French clowns and whatever--that's pantomime, which was also an accepted form in Roman theater, but which was basically striptease. In Attic theater mime was looked down on. Only a few fragments survive but it seems to have been a short interlude between plays, and were basically short burlesque skits, using farce and slapstick rather like you'd see in old American vaudeville performances. Sometimes they could get quite lewd and baudy (sex jokes and striptease seem to have been rather common) and unlike Old Comedy, which had in its core a very serious social and political message, they were just stupid for the sake of cheap laughs. Mime under the Principate grew to become an extremely important theatrical form, eventually more highly regarded than comedy and tragedy combined. We don't have much evidence for what was in them, but it seems to have been lewd, obscene, and grotesque, although the high form of Roman mime seems to have generally also been somewhat sophisticated, at least in form.

Now, what other evidence for humor do we have? After all, both these forms were largely borrowed from Greek originals. Well, plenty of other authors have humor. Catullus was a master of the epigrammatic style, and his poems are riddled with little jokes all over the place. We mustn't take Catullus to be indicative of all Romans or even Roman poets, since he was a master of his style, but he's rather fond of oblique attacks, jokes about impotence, and (he loves these ones) all kinds of different synonyms for the word "whore." Martial, also an epigrammatist, is also quite biting, but a great deal less lewd, mostly relying on his sharpness and wit to insult (he's not quite so interested in humor as he is in irony). Ovid also has a lot of jokes which, as you'd expect based on the type of work that he mostly wrote, revolve around sex a lot. And of course there's the great satirist Juvenal, but unfortunately for our purposes Juvenal isn't interested in making actual jokes, but in attacking his society in a way that ends up incidentally humorous (or, often, doesn't end up humorous at all). But Romans, like everyone else, told jokes regularly so we have a huge number of examples outside of authors in whose works we'd expect jokes to appear. Cicero likes to make jokes in his speeches quite regularly, although they're rather dry, usually not very funny, and mostly attacks on his opponents. Caesar doesn't make any jokes in his surviving writings (after all, they were dispatches to the senate, and it would be wildly inappropriate to joke around) or in any surviving biographies of him, but during his triumph in 46, B.C. when his soldiers were allowed to sing whatever songs they wanted (as was customary during triumphal processions) the legions sang about his notorious womanizing, joking that the Romans would have to lock their women away and telling rather nasty jokes about how he lost his hair (there were a lot of jokes told about Caesar's baldness--since he was very touchy about his lack of hair--and in Roman society premature baldness like Caesar's was often associated with promiscuity). Another thing is that we can see in inscriptions and graffiti an awful lot of insults and humor, often linked. The third paragraph of my writeup here goes into some of that. Once again the Pompeii graffiti are very fond of sex jokes, but one thing that they really enjoy for some reason are shit jokes. One thing the Romans seem not to have enjoyed much are fart jokes (unlike the Greeks, who found farts absurdly funny)--the only ones I can think of off the top of my head is in Seneca's cruel satire of Claudius, the Apocolocynosis. In that there are several fart jokes (sometimes paired with diarrhea jokes) about how Claudius' soul "bubbled out" from his body (and at one point Claudius lets a giant one rip) and so forth. It seems likely that this was considered particularly cruel, since most of what Seneca writes in that work is extremely mean-spirited and far beyond just satire. Another thing we don't see in abundance--even at Pompeii--is penis jokes. While the Romans seem to have been obsessed with making fun of people for submitting to degenerate and degrading sexual acts, they didn't really seem to make fun of men's penises much. This seems a bit odd, since it's one of the favorite jokes in Aristophanes and constantly appears in Greek literature, but there isn't very much evidence for it in Roman literature, although Catullus seems to find testicles somewhat amusing.