1. Pokemon
I've got a couple of friends who are into anime. Whenever they ask me what my favorite series is, I have to ask whether or not we're counting Pokemon in the question. Enough said.
2. Digimon
Meh. Never really watched it... looked like little more than a knockoff. I've heard decent stuff about a few of the seasons, but never enough to get into the broad series.
3. Sailor Moon
I'll preface this by saying I've only seen one episode of Sailor Moon that was aired by Kids' WB as a trial of sorts years ago. It could be interesting, based on its execution. I'm not averse to maho shojo, so that's not a problem. The people I know who have experienced the series are split on whether or not it's any good.
4. Yu-Gi-Oh!
Complicated question. Yu-Gi-Oh! was fun to watch in America. I've heard the dub was even better (particularly with the first season, omitted in the U.S.). The manga has also been great; I've read through volume 15 of the Duelist saga.
GX, though, is terrible. What little I've read of the manga seems a bit better, but despite a little more seriousness and wit it's still nothing like the original. Totally lost the original edge, with games having more significance than being just games.
Haven't seen the new series, 5R or whatever it's called. The games, on the other hand, are surprisingly lacking. They can't seem to make a game that both follows the rules of the TCG and has a plotline. As much as Duelists of the Roses might be fun, it's not the real game.
5. Naruto
This series has really, really grown on me. I've been a little stymied by the infamous filler arc, but it's still quite good. Great characters, surprisingly rich plotline, and a basic concept that just blends it all together without being overdone. (For a counterexample, see Bakugan Battle Brawlers. Ugh.)
My only complaint is that the series is too long for buying the anime or even the manga to be anywhere near affordable. Still, that doesn't detract from the fun of watching it on cable.
6. Dragonball
I haven't watched as much of this as I'd like to have seen, but what I have viewed has been good. If this series were released now, it'd be totally stereotypical. Knowing when it started, though, it's easy to see why it set the standard for the supernatural fighting genre in anime and beyond.
7. Super Mario
Never seen the show. The games, though, are classic. I really don't need to say anything beyond that, other than mentioning that Super Mario RPG is one of my all-time favorite games. (It needs to be re-released for the DS.)
8. Final Fantasy
I assume you're talking about the series and not just the first Final Fantasy title. Great games, tremendous soundtracks. I favor FFIX, myself. Some of the best music I've heard in any FF game, and a fun storyline and characters to match. Not to mention the skill-learning system, which is the best of any FF game (save, perhaps, for FFVI).
9. The Sims
Never actually played it myself, though it looks like an engaging time-waster. My main problem with this series is the number of new versions and expansion packs that systematically drain your real-world wallet.
10. Zelda
Another classic series. Link to the Past will always be the best of the series in my book, though the 3D adventures have their merits. I tend to get stuck in numerous places in the new games, unfortunately, and it's not the kind of genre where I feel I should need a strategy guide. My gamer pride gets in the way.
11. Crash Bandicoot
This used to be a good series, but it seems like it kind of fell apart at the seams. The whole thing just kind of lost its spark and became... lame.
12. Tekken
I'm not really much for fighting games, so I don't have enough experience to form a solid opinion. That said, Tekken is one of the top games in the head-to-head fighting genre. It seems like it's lost some ground to the Soul Calibur series, but it's still worthwhile for anyone into that class of games.