This is the heart of role-playing. The player adopts the role of a character and then guides that character through an adventure or campaign. The player makes decisions, interacts with other players, and, essentially, "pretends" to be his character during the course of the game.

So, within the game, the characters can be crawling through dank sewers underneath an abandoned city, being bitten by giant rats and contracting fatal diseases. But, in reality the players are sitting in a warm room, sprawled over the furniture, supping on a can of Old Speckled Hen, and munching on a ham and tomato sandwich. The players probably have pencils, character sheets or file that contain all of the relevant information about the character they are playing, and loads of dice. On a table you'll probably find painted lead figures that represents each of the characters and other characters that are with the party that are controlled by the gamemaster. The figures give a reference to how the party is ordered in the sewers. The figures are placed on paper that has a rough outline of the section of the sewer that they can see. The room is almost certainly in a mess.

What's the goal of the game? You might just as well ask what is the goal of real life. Ultimately, there are no winners or losers, you play the game to have fun. As the characters progress through the campaign they get better at what they do, and they try to fulfil the goals that they have set themselves. Each adventure has a goal, to rescue a princess, defeat an evil magician, or to win a space battle. If the gamemaster is devious, the goal maybe unobtainable, but there maybe a hidden goal that can be achieved. A campaign, is a series of adventures that has some ultimate goal, and can take years of playing to achieve, or die trying.

I'm going to break the question into two other questions:
What's a Role Playing Game?, and So What's D&D?