Dungeons & Dragons is a role-playing game, so you play the role of a character in a game world controlled by the GM (the Game Master, or in D&D they're also specifically called the DM or Dungeon Master). The GM is the person who describes the world and what all the characters in it are doing, except the players control their own characters. The goal of the game is for the players to complete the challenges created for them by the GM, and of course for everyone to have fun while they do it.

Each character has a set of particular abilities that you choose for them out of the options in the rulebook. Dungeons & Dragons is a medieval fantasy game so the options are stuff like Wizards and Paladins (Knights) and Druids and stuff like that. The most basic choices for your character are their race, background, and class: so you could choose to be, for example, a Human Farmer Fighter (a class that uses weapons) or an Elf Scholar Wizard or a Gnome Criminal Rogue (Thief), out of the many, many possible options. You also have stats: Strength, Dexterity, Constitution, Intelligence, Wisdom, and Charisma.

Whenever you do something important, you will most likely have to roll dice. The most common die to roll is a d20 (a 20-sided die). Rolling high is good and rolling low is bad, usually: if you roll high enough, you succeed at what you're trying to do, and if you don't, you fail (which doesn't mean you lose the game or anything, just that you fail at that one particular thing). Depending on your character's stats and abilities, you might have a bonus or a penalty to the roll, which is why you have to choose your character's abilities and stats based on what you want that character to do. For example, if you want to play a character that uses a sword, playing a Fighter and putting a high number in your Strength score will give you bonuses to using swords, while playing a Wizard wouldn't.

The game is designed so that the GM creates a story, but the exact outcome of the story will depend on what the players decide to do with their characters and some amount of random chance with how the rolls go. The most common stories in D&D are usually along the lines where characters play adventuring heroes (more or less) who go around slaying fantasy monsters and gathering treasure, but you can play a game that's about absolutely anything, really.

Does that all make sense?