Resident Evil 2
If you have not played Resident Evil 2, then you should definitely check it out. It's a straight up classic. The game picks up after the events of the original game, taking place a few months later. We find out that the whole town became infected.
The graphics have not held up very well.... but I have recently played through the game and it doesn't matter. The flow of the story, puzzles, and action is more consistent than the first game.
What gives this game 'Classic' status is the ingenuity of how you can play the game. The game comes in 2 discs, one for Leon, and one for Claire. Playing through the game with a character will give you a story distinct enough to make it worth playing through again with the other. But the real beauty of it is that when you, let's say beat the game with Leon, you save your game and now get to play through the game as Claire using disc 2... but in 'Scenario B'. You won't get the 'full' story of the game until you play through this scenario. The path you take is totally different, you actually get to see what Claire was doing while Leon was making his way through the police station. Anyway, this feature alone makes this game unique and worth playing through multiple times.
The characters are bad ass too... the main boss - William Burkin - is classic. The game is very cinematic, and today you can really see how ahead of it's time it was when it was first released. I mean, check out this cutscene. It makes Burkin look like a sav, especially since you know you'll end up fighting his ass later in the game:
Like I said, this game is a true classic. Most people that care have already played this, but just in case you haven't you should give it a try. The best version to get is either the PC version or the GameCube one. They have better looking graphics than the Playstation original. If Capcom had remade this game like they did the original, oh man can you imagine that....