If you want to learn more about Dungeons and Dragons check in on the Table Top Gaming thread on the forum! It's green-yellow in color near the bottom of the categories.
"D&D is a game of your imagination in which you participate in thrilling adventures and dangerous quests by taking on the role of a hero- a character you create. your character might be a strong fighter or a clever rogue, a devout cleric, or a powerful wizard. With a few trusted allies at your side you explore ruins and monster filled dungeons in search of treasure. The game offers endless possibilities and a multitude of choices- more choices than even the most sophisticated computer game, because you can do whatever you can imagine." -D&D Players Handbook 3.5th edition
Sounds great right? That's because it is, although unlike the before mentioned computer game a D&D game is only as good as It's Dungeon Master (aka Game Master) and the sum of It's players. So that's all well and good but what really is it. Well...
"It's part acting, part storytelling, part social interaction, part war game, and part dice rolling" -D&D Players Handbook 3.5th edition
I'm sure you are getting tired of quotes so let me elaborate on what the writer meant. While D&D involves imaginative scenes, interaction with local NPCs, and cunning use of skills, It also uses an in depth grid based combat system like a turn based strategy. Which personally is my favorite part as a DM although mostly because I'm bad at roll playing NPCs.
So how does one playeth the Dungeons and Dragons? We'll I suppose you'll need two things, a players handbook of whatever edition you are intending to play, and a Dungeon Master. I can give you the first one in a PDF form but the second one is harder to find. If there is enough reception here on Tek Syndicate I might consider trying to give Roll20 a go. Roll20 is an online tool that allows players to play D&D as if they were together with a battle grid, layer tools, sound system and everything.
Oh and you'll need one of these
A character sheet and some dice. Although if you are playing with a group online I suppose an interactive PDF is better.
If anyone is trying to find an interactive 3.5e character sheet I have one
But you haven't explained dice yet!
Right I suppose I got a little off topic. Dungeons and Dragons is a d20 based system. This means that the deciding factor for actions in most cases is a d20 (A 20 sided die) with relevant "modifiers" added to it. In action It's pretty simple and is practically the standard for table top RPGs other than the d10 system that role playing games like World Of Darkness uses.
Want to convince the panicked crook to lay down his weapons? Make a Diplomacy or Intimidate check using a d20+skill ranks+ability modifier. Then the DM will compare that number against the Difficulty Class DC of the check to see if it succeeds, fails, or fails spectacularly.
Combat is It's own very indepth system that involves turn based combat who's order is determined by a d20 roll with "Initiative" added to it. Than the turn is comprised of a number of actions that can be performed, which there are many types of actions that can be performed.
Fun fact each turn in considered to be roughly 6 seconds in game