I really enjoyed computer games growing up and it's a hobby I still indulge in during my free time.  Because of that, I've decided to maintain a list of games that really stood out to me over the years.  I'm not really a reviewer, although several of these are respected as classics.  All I really do here is try to bring attention to games that have both been enjoyable and have stood the test of time.

Also, keep in mind that this is not an attempt to create an exhaustive list.  I have to be pretty selective about what I play, so I usually limit myself to adventure or role-playing games since they're what I've traditionally enjoyed.  This is primarily because I've always enjoyed games with a great narrative, and this is a primary component of adventure/RPGs.

Quest for Glory Series
This series is one of Sierra's classic adventure games.  I've played most of the titles made by Sierra but this series was my favorite by far.  In particular, I was a big fan of the second one because it was the longest and really provided the most freedom.  As far as story goes, they were all very good.  The twist with this series was its use of an underlying RPG system to break up the quirkiness inherent in the adventure game genre.  It made the game a lot more immersive.  The only problem I have with the series is that the last game seemed thrown together rather quickly and didn't have the same feel.  That isn't to say that it was a bad ending, just that you should go ahead and play it last.
Betrayal at Krondor
I know I'm not alone in this one because it was Game of the Year almost everywhere at the time.  This game is based on a series of books by Raymond Feist.  There is even a novel based on the game written by Feist.  It is quite possible that this is the first first-person 3D RPG that I've ever played.  The 3D engine shows its age but the story and execution are excellent and no amount of technology can take that away from it.
Star Control Series (1 & 2)
The first game is a sort of action/strategy hybrid where the second adds a lot of adventure/exploration elements.  The first game is a classic, but the real treat is the second one which has more of an adventure/RPG feel with one of the best narratives I've ever encountered in a game.  I haven't actually played the first game in a while, but I'm constantly pushing people who like video games to try the second one, which is available for free here.

Also note that I really encourage you not to play the third game.  While SC2 is one of the best games I've played, SC3 is one of the worst.
Warcraft
I'll be honest, WC2 is the only real-time strategy game I've really enjoyed.  I've been trying to get into Starcraft, but it just doesn't have the same feel.  I enjoyed WC1.  I know it's strange, but I feel like the RPG elements were really a step back in WC3. 

As far as story goes, the original games were all great and lead nicely into World of Warcraft, which is one of the most satisfying online experiences I've found.  I tried playing WoW when it first came out, but didn't really appreciate it.  Recently, I've gone back and have gotten a lot more out of it.  I think finding a good group of people your first time out really makes all the difference.
The Elder Scrolls Series
This series is really interesting in that they seem to take the opposite approach of most other successful RPG developers.  Bethesda tends to add a tremendous amount of content to try to make the world as believable as possible.  Unfortunately, this tends to come at the cost of immersiveness. In particular, the characters and side-quests can be somewhat two-dimensional.  That being said, the game mechanics are fantastic, and you really get the impression that they're converging on how to make a truly believable world.  Just keep in mind that these games are beautiful and complex, but you might want to just stick with the main story line otherwise it can become long and repetitive.
Prince of Persia:  The Sands of Time

This is actually the only game in the series that I've played.  I heard bad things about the second game and consequently never made it to the third.  I can tell you that this one was very good and I fully intend to play the others.  It's basically an action/puzzle game with a story that is very entertaining and funny.
Castlevania
Like most long-running series, this one has its hits and misses.  Basically, I'll give you my favorites:  Simons Quest (CV2) and Symphony of the Night.  This isn't to say that I didn't enjoy the others, just that these two were definitely worth the time.  They were also the ones with the most extensive RPG elements, which may be why I liked them a bit more.
The Longest Journey

This series currently consists of two games.  The first one, The Longest Journey, is widely acclaimed as one of the best adventure games ever made.  The puzzles are solid, the characters are really likeable, and the story is excellent.  The first game is a completely stand-alone adventure and is just the right length.

The second game in the series, Dreamfall, is a little harder to endorse simply because it didn't have an ending.  The game is actually quite short and doesn't come to a satisfying conclusion.  The developers intend to release the sequel online in an episodic format, but I don't believe it's under development yet.  That being said, Dreamfall is quite good, just incomplete.