Most of the show's humour comes from the rivalry between Kyle and Cartman. In the earlier seasons, this was as simple as Cartman mocking Kyle's religion and Kyle calling Cartman fat - however, creators Parker and Stone have really run with this over the years and for me, the peak of the rivalry is in the episode "Tonsil Trouble". Cartman has his tonsils removed and during the operation he needs a blood transfusion; turns out the blood was contaminated with AIDS virus and rather than being sympathetic, Kyle finds this utterly hilarious (as you would). To teach him a lesson about AIDS not being funny, Cartman sneaks into Kyle's room when he's asleep and injects Kyle with his blood - giving him the disease too. Bravo!
In the first few seasons, Kenny was killed off in every episode, which became a bit of a gimmick. A fair portion of the fanbase would just tune in to see how they'd choose to kill off the kid in the parker. Kenny is slightly more rebellious than the others and seems more educated in the adult word. In the earlier seasons, Kenny was the one to explain most sexual innuendos that occurs were explained to the other three characters.
In the most recent season however, Kenny has received a whole heap more depth. In the episode "Mysterion Rises", all the kids around South Park are play superheroes, and it is revealed that Kenny has an actual superpower - the ability never to die. He explains how he feels and remembers every death but then just wakes up each morning in bed with no one having any recollection. Being a fan of South Park and seeing all the ways he has died, it's a bit harsh and nightmarish to imagine that this 8-year old has legitimately died all those times - it's really no wonder why he's the kid that does all the drugs (and, cat pee, in the episode "Major Boobage").
There are two other characters who have really come into their own in recent seasons. Butters Stotch and Randy Marsh. Butters is a loveable person; he's just so gosh darn cute and by far my favourite character. His little nervous quirks and innocent naiveté makes you want to give him a hug; however, these mannerisms certainly don't stop him being the subject of numerous practically jokes. In "AWESOM-O", Butters is convinced throughout the entire episode that Cartman is a gift robot mailed to him from Japan. Oh, loveable Butters.