Resident Evil: Director's Cut
Resident Evil was a game that defined the original Playstation. It was one of the first three dimensional games, along with one of the first horror games. It introduced a new concept not fully experimented before: "survival Horror". Resident EVil puts you in the middle of a viral outbreak that has turned everyone into zombies.
Although Resident Evil 2 blew this game away, there hasn't been an RE game that has really captured the grittiness of the original. Resident Evil was remade from the ground up for the Game Cube, and it enhanced it in every way possible.
So why should you play this old ass game, when there is a superior remake of it? Just for the experience. It allows you to appreciate things a bit more. You'll see innovative elements, things people liked (like a male and female starting character, big guns), as well as flaws and areas that need much improving (game saves, voice acting, puzzle arrangements). The Director's Cut version included a new mode that rearranged item locations and zombie placement (along with a few surprises) so things would feel fresh to veteran players. It also included a demo for Resident Evil 2, which I remember was bad ass and couldn't wait to buy the game.
Resident Evil has a pretty good story, which makes you hate the movie even more. Why didn't they stick to the original? Anyway, this game is a classic that laid the foundation for the rest of the series.