There are actually several styles of D&D out there, so let me break it down a little bit:
1. One of the forms of D&D is played with miniatures and elaborate setups of battlefields. The miniatures used represented small armies (called "units"); thus, one miniautre often represented many creatures. Moves are declared (i.e. My orcs will attack the elves with swords), each unit's order of play established by random dice rolls (called "initiative"); on each unit's turn, the miniatures were moved accordingly and random dice rolls were used to determine the outcome of the move (i.e. a dice roll would be used to determine how many orcs struck the elves, and how much damage was done to the Elven units). This style of D&D is still played and is sometimes used with Advanced Dungeons and Dragons (see below) to resolve mass combat situations. When the 2nd edition Advanced Dungeons and Dragons (AD&D) was introduced, the rules for this style of play was rewritten and introduced as "BATTLESYSTEM".
2. D&D, the original, was played according to books. Characters had multiple choice actions that they could take. This is an outmoded and outdated form, replaced by AD&D (Advanced Dungeons and Dragons). I don't think there are any more original D&D players out there any longer, as no one is writing new adventures for this form.
3. AD&D is the most common form of Dungeons and Dragons. It is also, in my opinion, the most fun. With AD&D, a character is generated using random dice rolls. The dice rolls determine several aspects of the character (Strength, Constitution, Dexterity, Intelligence, Wisdom and Charisma). From there, a race was chosen (normally Human, Elf, Dwarf, Gnome, Halfing or Elf, though some Dungeon Masters do allow alternate races), and class is chosen. For Class, there are four major groups: Warrior, Wizard, Priest and Rogue. Warriors are your fighters with the most skill in combat but can not use magic, Priests are servants of a diety who use divine magic and have some combat ability, Rogues are your thieves, assassins and bards with little or no magic ability but special stealth and pilphering abilities, and your Wizards are spellcasters who use spells from their own knowledge and study (as opposed to the Priest whose spells are granted to them by a diety). When all of this is completed, the Dunegon Master (DM) prepares an adventure, often creating maps, towns, other characters (played by the DM), etc. The characters are then dropped into the situation. With AD&D, the characters have a full range of actions and options, unlike D&D or BATTLESYSTEM where the options are limited. Thus, a character can run, try to bluff its way of a situation, try to join the enemy, hide, commit suicide; options are limited only by the imagination of the player, the abilities of the character and a few broad limitations set by the rules and/or DM. The most important aspect of this mode of play is called "role playing". You, as the player, get inside the mind of your character and make decisions based on your character's background, experiences, race and other factors. Thus, the characters in a AD&D campaign can be very rich. The successes and failures of actions chosen by the player are most often determined by die rolls.
As there is a lot to this game, I don't know where else to go with explaining how the game is played. That is the skeleton of it; we are more than happy to answer any questions you may have.