It's really just an advanced form of the high-score screen on many video games. In theory it's an indicator of skill in a game or if that isn't a huge factor just the sheer amount of time played.
I like achievements because they give me secondary reasons to spend more time playing and exploring games. Provided the game is worth playing in the first place achievements keep me playing longer and increase the value of the game.
Most single player games have a definitely end-point to them. Players are working towards the end-point as a goal. Why not a few road-stops along the way?
Multi-player achievements are just more troublesome because of the unpredictability and difficulty of real people, making them potentially a lot more frustrating to achieve.
> Achievements are worthless anyway. You should be playing the game for its gameplay, not to win some little icon in your profile.