Stand up comedy is an artform that is incredibly difficult to master. It usually involves one person on stage, with a microphone. In a set, typically lasting between 30 and 120 minutes, standup comedians do all they can to make the audience piss themselves laughing.
Standup comedians generally come in several flavours:
Prop comedians use props and gadgets on stage (Carrot Top, Gallagher)
Observation comedians try to look at events or situations, and try to observe the irony in it. The art of this form of comedy is to see what everybody sees, but explain it in a funny way. Jerry Seinfeld is the most famous name who does this.
Topic comedians do much the same as observation comedians, but they also tend to focus more on one particular topic. They then use quotes from other people and observations on this topic to make the audience laugh. Think Jay Leno.
Character comedians, in stead of telling people about things, take on a different role on stage, and use self irony to make fun of the character they themselves are playing. Tim Allen has had a few legendary shows when it comes to this.
Physical comedians are pretty much a mix of the above, but in addition to this, they are very physical on stage, doing silly stuff. Think Monty Python and Jim Carrey