In the beginning, writers created motivation for their characters and, having done so, gave birth to stories.
Okay. Enough of the scriptural prelude.
While a lot can be said about characters and characterization, I would be remiss if I didn't start with the protagonist. Many people refer to the protagonist as the good guy, but I don't really believe in good or bad people, especially when it comes to literature. The best characters aren't stock or fill-in-the-blank characters; they are ones that make decisions that may or may not have adverse effects on others. It's kind of Zen-like when you think about it: characters define themselves by what they do. As such, a protagonist might best be described as simply "the main character," the character who enters the hero's journey or seeks to find something. And in keeping with the disposal of good and bad, the antagonist would simply be the character with a motivation that runs directly opposed to the motivation of the protagonist. These conflicting motivations give rise to conflict, which is what gives rise to plot (but that's another discussion altogether). *Note: Conflict is not just the effect of character on character motivation; it could arise from a number of factors, including nature, self, or a higher force.
If a character goes through changes, or is dynamic, he is considered round. That simply means that the character at the beginning of the story has evolved by the time the resolution comes. More than 90% of the protagonists in literature are round.
Flat characters are the opposite of round characters in that they are static and don't change. Hollywood tends to play with this type of character in movies that involve people with mental disabilities. While a mild argument could be placed on Raymond from Rain Man being round, he's actually much more of a flat character. As a result, flat characters tend to be accompanied by round characters. Just look at Forrest Gump. He had Jenny and Lieutenant Dan to balance him out. And Lenny had George in Steinbeck's novella Of Mice and Men.
Stock characters are best described as characters that, once they are seen, the reader automatically knows what role they serve. Just think of cowboys with white and black hats.
A final category of characters are archetypes, which, in the wrong hands, can become stereotypes. Each culture has its own archetypes. With African-American literature in particular, there are a number of them: the sambo, the mammy, the tragic mulatto, the bad-ass, the buck, the militant, the sapphire, and the list goes on and on. Just one look at a TV show like Good Times, and one sees many of these archetypes on full display. At one point in time, archetypes were original character types, but history has shown us that readers tend to respond to characters whose traits are recognizable in some way, so now archetypes straddle the line between being rehashed character types and stereotypes.
While a lot more could be said about characters, the main thing to remember is that they are representations of human beings (which is a pretty heavy notion when you think about it). Everything else is just a way of explaining how a character functions within a story.