Family sitcoms, at their core, are just stories like anything else. So, as with all stories, they have to have a good plot, good characters, and an emotional connection with the audience. Of course, they have to have "adult" humor to appeal to their target demographic, but I think the most prevalent factor of the adult sitcom is satire, particularly social and political satire.
Let's compare a few:
Simpsons- As Zeph stated, 90's Simpsons is pretty much gold. Doesn't get any better. Awesome satire, humor and some genuine dramatic moments that don't feel forced. Modern Simpsons is just meh. Everything about modern Simpsons feels so artificial and forced. The plotlines are often convoluted and the characters don't act the way they should. Occasionally they may have a relatively good one (I liked last year's Christmas special), but the show has really worn out its welcome.
Family Guy- You know, even early Family Guy wasn't that great. Yes, it had the "outrageous" humor that is pretty funny the first few times around, but it was also really facetious in how it handled its plotlines and morality. But still, it wasn't bad. Modern Family Guy has now, of course, devolved into an unfunny, unpleasant, mean-spirited, sexist, racist mess.
American Dad- For a while it looked like this was the only passable Seth MacFarlane show left, but even this one seems to be going to the dark side now. American Dad had some really interesting stories and characters and (shockingly) managed to be relatively reserved in its political commentary. The humor was also pretty darn good and they took the show in some interesting directions (namely, Haley getting married). But now, as I said before, it seems to be taking a turn for the worth, mainly with Roger degenerating from a spoiled narcissist to an amoral psychopath.
Home Movies- On the flipside, this is a very character-driven show. There's not a lot of satire here, or if it is it's very understated. Rather than caricatures, the characters in this show feel like real people and although they don't usually get into any wacky situations, the humor in fact comes from how human they are. They're self-centered, cynical, but they also care for each other. And they fail. Oh, they fail.
South Park- South Park had a big impact on television, but I fear that a lot of writers and producers took the wrong lessons from it. Somehow people got it in their heads that raunchiness=quality, but in South Park the raunchiness served a purpose. It was never just about profanity or gross-outs, it was a tool to emphasize and illustrate their message. Matt and Trey are all about political and social commentary and they do it mercilessly, spelling out exactly why this topic of the week is dumb as hell. They continue to do this today, although I don't think it's quite as good as it was in its early years. The show used to have a certain charm and timelessness that I think the show is lacking now.
King of the Hill- How this show lasted so long I'll never know, but I gotta give it props for staying fairly consistent in quality throughout its whole run. This show is all about satire, and its got a really understated humor that may not appeal to everyone (it actually took me a long time to get into it). No, it was never as good as Simpsons, but I'd say it was a respectable sitcom.