Middle EarthBoard

The way it is played is somewhat difficult to explain. Mostly roleplaying, in some of the same ways that you see here in NationStates. The way it was pitched to me, in my youth, was "It's like an RPG video game, but you can do absolutely anything!" And that is mostly true, because it's only limited by the participants imaginations, the skills of your Dungeon Master, and some of the rule sets that give boundary to the game.

For instance, if you encounter a river in a video game, chances are you will have very limited options: cross, fish, perhaps find a boat. But in D&D, you can decide that you would like to build a dam on the river made from the slain bodies of your enemies, then declare war on the local beaver population that shows up to claim the dam as their own. And your DM (Dungeon Master) will attempt to facilitate such a story and goal, as best as is feasibly possible, given natural limitations of the world and rules.

That figure, the Dungeon Master (DM), will be quite important. They have to weave the story that the party of players will adventure in. They can either strictly follow a preset/prefabricated adventure, with a built-in story and stock characters, or they can do some preliminary work crafting a world with a customized mystery and unique characters. They're like the narrator, if the narrator was also creating the tale as he went.

The other players form the adventuring party that will play in the world. You can simply start with one of the available prefabricated characters, or create your own. This is often a favorite part for players (it is for me). Choose a race (Elf, Dwarf, Human, Halfling, Gnome, etc). Followed by a class (Barbarian, Wizard, Ranger, Cleric, Rogue, Bard, Paladin, etc). Moral alignment (Good/Evil, Lawful/Chaotic). Then you try to flesh out the back-story, as much as you're willing.

These are some fun sites explaining such things:
http://dnd5e.wikidot.com/
http://easydamus.com/alignment.html

With your character all set, you then follow the DM's story, responding to obstacles and choices in as close to as "realistic" a way, given your character's traits (race, class, religion, moral alignment, history, background, etc). For instance, if the DM throws a monster in front of you, the party may choose to attempt to slay it, or flee it, or befriend it, or trick it, or sing it a song and try to teach it to dance. But, whatever action is taken, the outcomes are decided by dice rolls, applied to the characters' stats and skills (Strength, Dexterity, Intelligence, Perception, etc).

Combat, frequently the most common encounter, will basically entail choosing your action (attack with sword, attempt to dodge, cast spell, etc), then determining damage/effect, via dice roll and stat check. But you would use your skills outside of combat, as well. If you enter a new town and walk up to a bartender and try to get information out of him, you will likely roll a skill check for your Persuasion. Players can do any number of things in a given situation, and frequently the result will come down to whether or not you can convince your DM that something is possible. A good DM will try to balance fair judgment with flexibility for facilitating fun.

Lastly, there are sort of two different styles of gameplay: Map and Grid, or Imagination. Basically, you can have a map, of varying degrees of detail, demarcated by a grid, with the characters represented by miniatures/figures that move along that map in a steady manner. This is more like a video game, sort of a dungeon-crawler. You can be more strategic, more specific about movement and placement. It seems to often resemble a strategy wargame.

The other style is more Theater of the Mind. The DM paints you a word-picture of the setting, your surroundings, what is happening, and progression is depicted verbally. This can be freer, more fluid, but also more chaotic and confusing. But it's the style I prefer. Though it does seem to require a more talented DM to pull off well. Often, a combination of the two will be employed. Narration for the more simple occasions, but maybe drawing a detailed map and grid for important battles.