That's a big question, but here's some starters.
D&D rules began as rules for a pen and paper style game that could be played live by a bunch of friends (i.e no computer needed! <img src="/ubbthreads/images/graemlins/smile.gif" alt="" /> ).
One would take the role of Dungeon Master and control the play.
The fights are all based on the rolls of various dice. I can't remember how many but they have different numbers of sides and not just the usual 6.
The idea is that a basic fight is made more complex and interesting by adding all sorts of 'modifiers' for chance to hit, agility of player or whatever - I can't remember the technical details. Some of your stats you keep, and build on as you go, but other parts of the outcome are determined by the dice rolls.
Similar things happen in many computer games - even non D&D ones but the calculations remain hidden under the surface, and the player only sees the end result.
The system is copyright so you may not find too many of the rule books on the net as the book is the bit you buy. I think you can also buy preset stories to follow. Not sure of all the details.
Anyway, the computer games follow the rule sets (with minor variations). However, you don't see all the dice rolls and don't need to know the rules to play. With some you can get the numbers to appear on screen if you're interested though.
With some (Like Baldurs Gate 2) you get a thick manual with all sorts of tables and charts to pore over if that's your thing. However, it's not essential to do so.
In general, the difference is less emphasis on hack and slash (although there's still plenty of that) and a bit more story, quests etc. Oh and more party oriented too.
