Some great recommendations already, I'll add these: (IMDB isn't letting my access my list ATM...)
-Scream 1 (been mentioned, first saw it years ago and it remains one of my favourites - don't spoil it for yourself though). I also like the second one...
-The SAW series (you'll like these if you like Cube - they're quite different though and more of a specialised taste, but you should like them if your into horror)
-Aliens (not strictly a horror movie, the original Alien is, but I actually prefer Aliens)
-The Thing (80's version)
-Jaws (technically classed as horror, but very tame by modern standards)
-The Ring (haven't seen it in many years, but I remember it being OK, apparently the original Ringu is better)
-The Mist
-28 Days Later
-Controversial one: The Blair Witch Project (I actually like it, but it's like Marmite, you might hate it)
Very good suggestions made by others, I envy the fact that you get to enjoy new horror movies for the first time, it gets harder and harder to find good ones the more you watch.