The Halo series is the default FPS. These are the games by which all of FPSs are judged. You need to own at least one of them.
On console, pretty much.
For shooters, I can't really agree with the praise given to Halo. I found the Halo games to have moments of genius interspersed with long stretches of suck.
While I wouldn't go quite so far with the wording, I do tend to agree, particularly in regards to Halo 3 and ODST.
Halo: Reach, however, I found to be entirely too awesome. If you intend to only get one Halo game, make it Reach.
Another great game to check out would be Bioshock (and its sequel). Really brilliant environment, frightening antagonists and a truly memorable and immersive story. Can't believe I forgot this one, actually. One of my favourite games evar.
Bioshock is truly amazing. Probably the best story in an FPS I've ever experienced. Bioshock 2 was good, but didn't quite measure up IMO (plus your character in 2 took forever to actually become durable in a fight). The Minerva's Den DLC for Bioshock 2 was great.
I suppose I should mention some cross-platform games... Previously, I was only sticking with 360-exclusives. You may already have these on your shelf, and if not, they should be fine on either system.
Darksiders is very cool action game, borrowing a lot from Zelda and God of War while also making a really cool history and world.
Castlevania: Lords of Shadow is also an excellent action game (again, borrowing quite a bit from God of War) but the story is... weirdly done - much as I love Patrick Stewart, having him give a short, badly-written monologue at the start of every single level was not the way to go.
The
Assassin's Creed series is a fantastic batch of games. AC1 is probably the weakest entry in the series, but the others are worth checking out, at the very least.
The new Batman games (
Arkham Asylum and
Arkham City) are downright amazing. Asylum's final boss is a letdown, but the rest is brilliant. And City is one of my favorite games of all time.
Brütal Legend is great fun with plenty of metal-infused humor and one of the most vibrant and original settings I've ever seen. Plus, it's got a great voice cast and soundtrack (if you're a heavy metal fan).
Ninja Gaiden II (and the PS3 Sigma upgrade) is one of the most blisteringly brutal, controller-busting difficult, yet viscerally satisfying games on the market. But only get it if you want totally over-the-top action and a serious challenge and have no problem with tons of gore and flying limbs.
Shadows of the Damned is a crazy, fun, creepy, (vulgar, raunchy,) 3rd-person over-the-shoulder shooter assault on Hell. Lots of extremely tongue-in-cheek humor plus some challenging gunplay and another amazing setting.