Role-playing games are simply games where each player takes on the persona of some fictional individual in some situation, and then attempts to act out what this person would do. Everyone's played role-playing games before, even if they don't know it. House, Doctor, and Cowboys N' Indians are some classics enjoyed by children. While children find these games great, adults get tired of them very quickly because there's only as many characters as there are players, which limits interactions, and disputes always arise along the lines of who yelled "Bang!" first.
To solve these problems, two new concepts are introduced. The first is a set of rules codifying who can do what to whom and when, so there won't be any arguments. Such rules are available in many varieties both online and in published books, each with their own individual flavor to them. The second element is a player who doesn't have his own character, but instead acts as referee. He not only gets to play the parts of all the extras the other characters meet, he is also responsible for coming up with the framework of the story in which the main characters are involved. This job is generally known by a title something like Game Master. Being the Game Master is a lot of work, as you have to invent plot lines, design locales, populate them with believable denizens, and respond when the players do something you weren't expecting, but it's also a lot of fun. Being the Game Master is lot like a combination of an author, a director, and an improv actor. It's also very rewarding when done right; watching players enjoy the adventures you helped craft can boost anyone's ego.
To play most role-playing games, you will need a little basic equipment. Most important is a group of players. Three to five, including the Game Master, is generally ideal. Second is a set of rules (usually in the form of a book), and most rules use dice in some form to add an element of chance. The last two things you'll need, and perhaps the hardest to get, are free time and an overactive imagination.