I never really got into the classic John Ford/John Wayne American-style Western. They're usually too stilted, too... clean for my tastes. A couple of exceptions, though:
Winchester '73 with Jimmy Stewart, and
The Ox-Bow Incident with Henry Fonda. The latter really isn't a Western in the traditional sense, it's mainly the setting that keeps it in the genre, but no one should go without seeing this one.
The Leone/Eastwood trilogy is a given, as should be Leone's Once Upon a Time in the West. I haven't yet seen A Fistful of Dynamite (Duck, You Sucker!), but it's on my list when MGM finally brings it out in a few months.
Beyond Sergio Leone's work, there are a lot of great spaghetti Westerns to see. I enjoy these because they're often bleak, grimey, and much more violent than their US-counterparts. To some, it might come off as schlock, but it's really no different than many action and horror movies from the same era. If you're just starting out, I wouldn't advise buying these outright, but definitely given some of these a rental to see if they appeal to you.
Django -- dir. Sergio Corbucci; Probably one of the best of the Italian Westerns to start with. Franco Nero plays Django, a drifter perpetually dragging a coffin behind him. He helps a woman being attacked by bandits, gets into a war of sorts with them, etc... Doesn't use the most original plot, but the results incredible nonetheless. The "big reveal" scene when Django finally opens his coffin is one of the greatest moments. This movie clearly influenced a ton of directors today.
The Great Silence -- dir. Sergio Corbucci;
Jean-Louis Trintignant is Silence, a mute gunfighter and Klaus Kinski is, as always, a fucking badass. Takes place in the dead of winter in Utah, which makes for some incredible visuals. This has an absolutly killer ending; be sure to check out the mind-bogglingly bad studio-sanctioned alternate ending on the DVD. It's awful to think that shit like that still goes on today.
A Bullet For the General -- dir. Damiano Damiani; Klaus Kinski is awesome here. Good political plot, although I think it runs a little long.
Great ending.
Compañeros -- dir. Sergio Corbucci; Franco Nero and Tomas Milian's characters make up a sort of oil-and-water team here. This is one of the more lighthearted entries in the genre and an absolute blast. Highly recommended!
