Replay value has always been a big deal, but perhaps now more then ever is it important to modern gaming. When a new game costs you 60 or more dollars after tax, getting your moneys worth is a big deciding factor. The first things people think about are usually...
1) How long is it? (How many hours to complete?)
2) How big is it? (Is it expansive, open ended, can I explore?)
3) How's the multiplayer? (Does it even have it and will I care to play it?)
More and more though, players are looking towards Achievements as well. (AKA Trophies for PS3 owners) Achievements are an artificial way to extend a games life without having to actually create and develop more content. Some game companies have incorporated them into their games with great success, while others have failed badly to make them worthwhile. An example of a simple achievement might be: Kill 500 of these bad guys. Then once you do, you get some points as a reward.
The value of achievements of course, is ultimately up to the gamers themselves. Most gaming companies actually keep track of an individuals total achievement points spread across all games, and give you a total point score. It's a way to show your experienced not only at certain games, but as a gamer in general. Depending on how difficult certain tasks are, you might get more or less points for completing them. For example, most games will reward you points for completing it. Most games will also reward you more points for completing it on a harder difficulty.
Some game companies though, go above and beyond with their achievements. Its very easy for some companies to just tack on rewards for doing basic things like finishing the game. However, few and far between you will eventually come across a game that you love to play and has fun achievements. Left 4 Dead is a prime example.
In Left 4 Dead, the achievement points are mostly rewarded for learning to do things that are actually important to be successful at the game. So you really WANT to complete the achievements. They will challenge you to do something very specific and once you've done so you're a better player for the experience. (Lets not forget, that they are actually fun which is important) You will be challenged to play the game again and again to get certain things done and because of how good the game is itself and how well the achievements are designed you'll want to. (Especially with friends)
On the opposite end of the spectrum completely are the achievements in the most recent Prince of Persia. They seem to "reward" you achievement points just for turning the game on. You will get the vast majority of the available points in a single play through, without trying. Once it's all over, you'll never care about bothering again. This game offers very little replay value comparatively speaking.
So what does all this matter? Well it's all about bang for the buck. Would amazingly fun and robust achievements have made Prince of Persia a better game? No, that's absurd. However, they would have added a lot more life to the game for those who did enjoy it and you simply cannot discount that. Achievements are now a major part of modern gaming. Game companies need to make sure they are crafted with as much care as the their games themselves. Because when they aren't, it shows.