regarding achievements, I like IN and have played it a lot, when I started playing Touhou, because I wanted to unlock all the Last Words and beat them. (however I can't fathom the mind of people who just go for the easy achievements)
perhaps, achievements can motivate the player to keep trying the game (perhaps the player has no experience in the genre, or no developed interest in the genre, or has difficulties on the game, or just has a lazy mindset, I mean, doesn't adapt or want to adapt himself/herself as well to the gameplay or improve at it). Take for example..
Ketsui Death Labelhaven't played the game yet, but the achievements seem to be "color-coded" according to their difficulty. How nice of Cave, to try to be welcoming to new players

(while we're still on the subject of achievements, Street Fighter IV's Time Attack/Survival/Trial modes could also be considered achievements; the Trial mode tries to teach you combos and show you what your character can do, while the other modes -- and the normal single gameplay as well -- allow you to practice your current knowledge and abilities, to an extent)
As for what motivates me to play, in the case of competitive games, is that I like to test myself and try to improve (and feel that I've improved). Games that aren't meant to be competitively played normally don't have a competitive playerbase, which you need in order to improve in said game.
For non-competitive games (but clearly not limited to), there are single or multiple elements that can be enjoyed: gameplay, storytelling (even if a game doesn't have a deep story to tell, it can have a background story, or make you feel relaxed, stressed, pressured, etc., depending on where you are in the game), immersiveness (stage + bgm + player character, for example), sense of accomplishment, incentive (reward) to advance in the game (some later stages would feel more fun for me to play, so I would want to get to them as soon as possible, but it's still important to make the first stages not too boring). Perhaps we could bunch all these factors (and others that I might've forgotten) together and name them "gaming experience".
here's something that I would like to show you people:
http://www.theoryoffun.com/theoryoffun.pdfGames can entertain, and exercise, among other things.. we like to be exercised, as long as we're enjoying it, it seems.
now, something that I wouldn't like to show you, but feel like it's necessary in order to try to explain the fun in trying to get the most of a game's intended gameplay.
http://www.sirlin.net/ptw-book/intermediates-guide.htmlwhat I said above (fun in trying to get the most of a game's intended gameplay -- or, trying to push the game to its limits, or yours) applies for non-competitive games too..
http://www.youtube.com/watch?v=Hgd3YcmIvYE.. the difference, is that the example above has some (lol xD) inhuman execution demands (air jumping), and unintended features (wall-zapping, air jumping, and other stuff you may read
here). Still, exercising your knowledge to perform better in a game, along with other virtues (willingness to take risks, reflexes, reading ability, adaptation, improvisation, patience, initiative, control, etc.) was at least part of the fun I meant to express, though the practical example (a Tool-assisted superplay) was merely for illustrative purposes, since most people here would be familiar with Megaman.