I'm not really sure how to begin...
I've played all the classic horror games, the Silent Hill games, Penumbra, FEAR, Condemned, System Shock, etc. Not one of them comes close, although Penumbra deserves second place and bears some similarities - it was, after all, made by the same team.
Amnesia isn't typical in any way. You don't fight, you have no guns or explosives. It's raw survival that drives the action. The elements that make the game so uniquely terrifying have more to do with what you hear and feel, and less to do with common gimmicks like monsters jumping out of the shadows (not a bad tactic, but one that grows tiresome over time). When you create a new game, on-screen text advises players to experience the game, rather than win it.
Before playing, make sure that you are in a dark room, adjust the brightness as per the game's specifications, and play with headphones. Since so much has to do with mood, and since the atmosphere is so compelling, it would be a disservice to the game and yourself to play it any other way.
The story itself is fairly simple - a man has lost his memory and is trying to put the pieces together. But there is something.... wrong. As you progress through the game and begin to see the whole picture, the sense of "wrongness" increases to a nearly hysterical level.
I enjoy watching good horror movies with friends, it might make me a bit sadistic but I do enjoy sharing their fear and giving them a bit of a scare. I can't think of a film or game that I would tell an adult friend not to watch or play. But in the case of Amnesia, be warned - this game is DARK. Some of the themes and scenes are extremely disturbing. It's the only game I've ever had to take breaks from because my nerves were so frayed I wasn't even having fun anymore.
Furthermore, if you're a parent considering purchasing this game for a child, please don't. There are some things you really don't want in your head, and some of those things could be potentially harmful to a child. Some of them could be harmful to adults as well, I suppose.
Anyway, if you love horror games and are looking for more, pick this up immediately - it's the best one yet.