Achievements are interesting - for some, the ego driven (the ePeen centric folk) it's about having the most points or icons so they can feel they've accomplished something. For others they play the role of both virtual sign post and milestone; a goal of "can you do this?" and a marker for when they do. For some others, it's a way to share memories of gaming moments, creating more of an emotional investment in the game.
Based on what I've seen, the vast majority of people who like achievements do so based on the ePeen factor. The only reason they do the usually random and pointless tasks required for achievements is so they can see their stats (in this case, icons and Gamer Points) increase. I don't particularly respect that mentality. I believe people should play games for the sake of gameplay, not grinding.