The concept itself is a good idea; the gamers get extra replay on games that would get boring pretty quickly, and the developers really don't need to put too much effort into making them. However, there are several common pitfalls that developers fall into when making achievements.
1) Pointless/"no effort" achievements: Achievements that you get for simply turning the game on or doing something that requires no skill SHOULD NOT be included in the game. They're called ACHIEVEMENTS because they should require some skill to do. This should also include achievements for passing the training mission.
2) Achievements you can cheat at: Nothing infuriates me quite like hearing some kid in Halo 3 try to cheat at getting the many online achievements. Offline achievements can sometimes be exploited too.
3) Spoiler Alert!: Getting achievements after completing a mission or chapter is ok, but developers need to make sure they pop up AFTER the next chapter has loaded. Nothing quite spoils the ending like finding out you've beaten the game (due to an achievement popping up) before you actually see the conclusion.
4) "High Volume" achievements: Ones like Seriously and Seriously 2.0 in the Gears of War series simply require a VAST amount of kills. Although this proves you've played the game a long time, in reality even the worst player would eventually get this achievement. Achievements that only prove that you've played the game for a long time usually aren't a good idea (Although it can be pulled off; see Rock Band 2's Bladder of Steel Achievement

). Getting all the flags in Assassin's Creed would take FOREVER without cheating.
5) Impossible Achievements: I'm talking to you, GRAW. There's probably like 20 total people who made it to the top of the online leaderboards.
6) Stacking Achievements: This mostly pertains to GH3, but can apply to any other game with the same system. If you insist of giving achievements for completing each difficulty (easy, medium, hard and expert all have their own achievements; honestly, I don't think you should get an achievement for beating Easy - something my 5 year old cousin could do), at least make it so that players like me who start at expert get all the completion achievements for beating the game. I am in no way motivated to go back and beat all the songs/levels in a game on an easier difficulty than I just did. If you beat it on Expert, why should you have to prove that you can do it on Easy?!