Trophies and Achievements are incredibly popular things, they add new areas of competition and bragging rights and can greatly increase a games longevity. People even become addicted to unlocking them and take an immense amount of pride in their Gamerscore or number of platinum trophies. Personally though, I hate them.
On one hand trophies can add to games, completionists are rewarded and masochists have something to brag about. But what about the rest of us? Personally, so long as I've done the main story of a game, and possibly a select few side quests, then I consider it completed and I'm done with it. In the past this wasn't a problem but now people can see that I haven't got 100% of the trophies. People think I haven't completed the game and that it looks bad, essentially many gamers are punished by the trophy/achievement system rather than rewarded by it.
Also if you were to borrow a game and then find out that you didn't like it that game is listed as being 0% complete and, quite frankly, looks messy and ruins the list of games that you have played by again making you look like a poor gamer. This shouldn't be how it is, you should just be happy knowing you've completed these games, haven't completed those ones and didn't enjoy the others. You shouldn't be judged for it by being given a big fat zero for others to laugh at.
My other major problem is that it causes some gamers to simply miss the point of the game. Instead of paying attention to the great story and awesome gameplay of a game like Uncharted 3, some gamers just focus on getting a platinum trophy. I even know of some people who refuse to play a game like Metal Gear Solid 4 because it doesn't have trophies. This is utter madness and it is bad for gaming.
Ultimately I'd appreciate it if the next generation of consoles did away with trophies and achievements altogether, some level of in game rewards might work as an alternative, the Metal Gear Series does this quite effectively in the first 3 MGS games but did admittedly lose track in the fourth. Only time will tell how things shape up with the next generation but hopefully at least Nintendo will keep their games trophy/achievement free.