I saw some sort of interview with the creators once, and IIRC they said that they loved Archie Bunker’s character and wanted a similar character for their show. They made him ten years old so the censors wouldn’t make them get rid of him.
Sweet. 
They also said one of them was to be represented by Stan and the other Kyle, but both decided they felt like Cartman.
Well, there was that one time he tried to start another pogrom against the Jews after seeing Passion Of The Christ…
This one time, at Fascist Camp. . . .
I think it’s because Cartman serves as a relief valve for the creators. It’s a habit that most of us have, when we’re really pissed off by someone, to want to hurl invectives at them which feature some characterisitic of theirs that we don’t have. We’re not bigots, it’s just what pops into our heads at the first blush of anger. We don’t mean any of it, and we’re in control enough of our emotions that we don’t allow ourselves to spew it out, but it would feel so good to lash out with every offensive thing we can think of. Cartman lacks that inhibition, so even the mildest offense gets him spewing slurs and other offensive things.
I dunno. Cartman hates alot of people but he seems to hold a special place in his heart for Jews and hippies. However I’m not sure if his anti-semitism is really an anti-Kyleism. To answer the OP, I think Cartman hates Jews to get at Kyle, hates hippies to rile up Stan (and Wendy), and hates poor people to rag on Kenny. But mostly he’s anti-semetic because it’s funny (since Stone is Jewish, SP can get away with making a joke about anti-semitism just like Chris Rock can say things about black people that would get a white comic run out of town on a rail) and also because Parker and Stone like to push the envelope of taste.
I think that’s the answer.
I remember a scene where Ike tugs on the tablecloth and makes the Shabbes candles fall and set his hair on fire.
Cartman: “She was young and needed the money”
Have to disagree with the statements that Cartman is anti-Jewish because Kyle is a Jew. I think he hates Jews (remember the Passion of the Christ episode where he dresses up as Hitler and leads the town people in chants of killing Jews) and since Kyle is a Jew, he rips on him. Cartman is foremost a bigot and thus treats people of those characteristics badly.
I disagree. I think they are true friends. That is, inseparable. They put up with each other’s bullshit that they can’t stand, yet still stay together.
Dude, it was last week!
I think Smiling Bandit has a pretty good point, really. Cartman doesn’t think Jews et al are inferior because they fit into some category; he thinks they’re inferior because they’re not him. And really, he doesn’t even think they’re inferior. We are, after all, talking about a kid who has to get his validation from a stuffed frog and a doll named Polly PrissyPants. His toys tell him that he’s handsome and cool and smart, 'cause sure lord ain’t nobody else going to say anything of the sort to him. At the heart of it, Cartman is sad and lonely and insecure, and he runs his mouth to make up for it.
And, of course, people who run their mouths to make up for personal deficiencies tend to go light on people who mostly ignore them and focus most of their efforts on people who needle easily. Most of the other kids don’t really seem to pay much attention when Cartman pulls his shit on them, but Kyle gets all worked up. That makes him the most entertaining target, by quite a lot.
The biggie, though, is that it’s funny. Assholes can be pretty damn entertaining when they’re being assholes to someone besides you, and Cartman is nothing if not a consummate asshole.
“Git…out…of…mah…flashback…you…gad…damn…Jew!”
I’m not sure he even knows what a hippie is, much less a Jew. 
Sure he does: “They say they want to save the world but all they do is smoke pot and smell bad!”
Again, Smiling Bandit has a good point. I don’t think Cartman has any idea what “Jew” even actually means. Case in point, the “I didn’t mean it, you’re not really a Jew” comment already cited. He’s like my dad describes himself as a kid. Daddy grew up out in the boonies, in a county where you had your choice of religions: Baptist or Methodist. He heard vague rumors of some evil, drunken people called Catholics, but he sure didn’t know any. And he was sure he didn’t know any, because the horns and tails would have been a dead giveaway. He was fourteen or so before he knew being Catholic just meant you went to a different church.
He’s not bad - he’s just drawn that way.
No chance. Cartman has frequently tried to kill Kyle, for one. Kyle and Stan tried to have Cartman sent back to the evil dimension when Cartman’s evil (good) twin came to this one. They encouraged the robot from the future to kill Cartman in the Trapper Keeper episode. Cartman refused to give up his kidney.
The link here is Kenny. Kenny and Cartman are friends. Stan and Kyle keep Cartman around because he’s Kenny’s friend and because Cartman is often useful. Cartman hangs out because he’s got no other friends.
Of course, this is ignoring the real reason is because Cartman being part of the group is hilarious.
He makes an excellent foil for the others. Like in the “Fourth Grade” episode where gets the guys to agree to do something (like mooning Miss Choksondik) and then they fail to follow through. It takes him a few times to catch on.
NotWithoutRage:
What’s funnier is the aftermath, when he’s sent to the principal, who agrees that the Hitler costume is offensive and racist, so she makes him, on the spot, a “ghost” costume which to us less-clueless viewers, is actually KKK.