I never really got into fighting games, mainly because I am not the type of person who does combos. I am a total button-masher when it comes to games like that, since I am paying attention to the action going on in the game, rather than the controller itself.
That being said, I have always liked Super Smash Bros. It requires skill to play, and, at the same time, does not require you to memorize sequences of buttons that you have to press ridiculously fast. It still requires you to think quickly, but it does so in a much more logical way. Also, I prefer to play games with a story or some type of achievement system, as they give me a sense of satisfaction for actually accomplishing something. Super Smash Bros. is perfect for me in this way too, with the trophy system and The Subspace Emissary (in Brawl). To quote Pokemon, "Gotta catch 'em all!"
I have not played too many other fighting games besides Super Smash Bros., so perhaps there are more franchises here that are for me. For now though, Super Smash Bros. remains my favorite fighting game franchise.