I'd recommend the Baldur's Gate series, starting at the first one. It shouldn't give you any problems on an old computer, and it will be a great introduction. The graphics are dated, but I don't think the character development, character interaction, or storyline has been done better in any later RPG game. Many of us who grew up through games like this, consider the BG series something of a gold standard for RPG's.

After Baldur's Gate (if you finish all of them, and that could take a year or more, depending on your free time!), you might try the first NeverWinter Nights game. It's easier on PC hardware than the NWN2 sequel, and there are some great user-written story modules for it.

There is a more modern type of "RPG" that's sort of a combination of classic RPG concepts and the first person shooter game, like The Witcher, Bioshock, and Mass Effect. There is some character development, typically involving good or evil paths through the game, and a storyline to follow, but you'll spend a lot of your time fighting through action sequences. Of those three, The Witcher is probably closer to a classic RPG (you'll spend at least as much time following plot developments as fighting) and it's very good. There is also Oblivion, which I haven't tried and I'm not sure how closely it fits the RPG genre.

But all these combination 3D shooter/RPG's will require a fairly modern rig for a smooth frame rate. My computer is a couple years old now (Athlon 64 3800+, 2 Gigs Ram), but with a fairly recent graphics card (GeForce 8800 GTS). With that graphics card update, I can run at least the Witcher and Bioshock just fine, with smooth frame rates, if not the most hardware-crushing recent games like Crysis. So you might be able to try some of these newer games with just a graphics card update, if you're not close to a full system replacement.

And then there's Mount & Blade, a terrific simulation of mounted medieval combat. I'm having a blast with it, but I'm not sure I'd call it an RPG exactly. You can develop your character and his/her mini-army of followers in several different directions -- raider, rebel, or loyal underling to a King. So there's that role-playing aspect. But there isn't any set goal or storyline, so it feels a little more like a sandbox medieval simulator than an RPG to me, but your mileage may vary. You might want to try this, since it's not as demanding on hardware as most of the recent first or third person 3D games. Try the free demo, if you're interested. But get Baldur's Gate too!