Family Guy, on FOX
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See also What a Cartoon!
This show started in 1999, and it was cancelled in 2003, after three seasons. Later it became really popular in reruns on adult swim, and FOX subsequently renewed it in 2005. Since then it has continued for a bunch more seasons, and so far I have no idea when it might end. I wasn't exactly a fan during its initial run, but in general I thought it was okay. And I did come to like it better on Adult Swim, and then I guess I enjoyed the new seasons more than I would have expected. I suppose I always thought it could sometimes be hilarious and clever, and other times just stupid, annoying, and even offensive. And that's still true, but... I guess I acquired a taste for it over time. And probably it has found its footing more, and gotten more clever than it was originally... but it was probably always more clever than I gave it credit for. It's still not something I love, but... yeah, it's okay. And some episodes are better than others, especially some of the specials. However... in season 13 (if not sooner), I decided the funny to stupid/offensive ratio shifted for the worse, so I'll probably watch it less often from now on.
Anyway, the main characters are Peter Griffin, his wife Lois, and their kids, Chris, Meg, and the evil genius baby, Stewie. And their talking dog, Brian (who is Peter's best friend, but he's a lot smarter than Peter). The best things about the show were always Stewie and Brian. Also I kind of like some of the flashbacks and fantasy sequences and stuff like that. Some pop culture references and all kinds of things... can be quite funny. Let's see, what else? The Griffins have some weird friends and stuff. Peter's friends include a cop in wheelchair, named Joe Swanson; a womanizing pilot named Glenn Quagmire; and Cleveland Brown, who is... kind of the token black friend. Um, and Adam West is in the show sometimes, as the mayor of Quahog, Rhode Island, where the show is set. And there are any number of other characters. There's rarely much of a plot, or... even if there is, there's at least as much time devoted to completely random stuff. I really wish I could think what else to tell you.
In 2009, Cleveland left Quahog to star in his own spinoff series, The Cleveland Show, but after four seasons, his show was cancelled and he moved back to Quahog. On an unrelated note, if you like Family Guy, you should also enjoy the webseries Seth MacFarlane's Cavalcade of Cartoon Comedy.