A Gentle Introduction to (Narrative) Role-Playing Games
RPGs in a Nutshell

Role-Playing Games are about acting out a story with your friends... where you make up the script as you go along! Depending on your preferences, it can be like a strategy game: carefully planning your moves and trying to become more powerful than your enemies. Or, it can be like a soap opera, with dramatic moments and a crazy cast of characters. It also has strong improv elements, like the TV show "Whose Line Is It Anyway?" (and sometimes is just as funny). First and foremost, RPGs are a regular social activity where you get together and have fun. The rest is up to your imagination.
There are many different types of RPGs (Live Action, Computer, Online, etc.) but we'll talk about Narrative ones, where collectively you build on a story. The gaming group gathers regularly for a session which lasts several hours, usually 2-8hrs depending on people's preferences and schedules. During a session, everyone is seated together in the same room, usually at a big table (narrative RPGs are also called table-top games). Sessions have their own etiquette, just like every other social activity. For example, whether you should eat munchies at the table or wait for a communal food break is a common point of session etiquette.

Who is who?
The people in the group are called players; one special person in the group is called the Game Master (GM). Every player creates a fictitious persona called their player character (PC). As a player, you are responsible for making all the decisions for your PC. Your PC can become rich, famous, and powerful if played wisely... or reach an untimely end if played recklessly. Exceptionally foolish players also receive the ultimate punishment: namely, endless ribbing by their peers for years to come. :-p
The GM plays every other creature in the world, which are collectively called non-player characters (NPCs). In addition, the GM has many other responsibilities. First, they are a window into the game world, describing what players sense and experience. Second, they typically initiate storylines, placing obstacles, puzzles, and rewards in the game world for the PCs to overcome, figure out, and earn. Third, they are a rules referee, interpreting on the fly which rules are applicable in any given situation.

The movie analogy is helpful when thinking about RPGs. The players are all actors, and the PCs are their characters in the movie. The GM is the producer (deciding the general direction of the film) and plays all the supporting cast and extras (everyone who isn't a main actor). But unlike a movie, where the plot has been decided beforehand and the actors are merely reciting their lines from a script, in an RPG the people improvise their lines and actions in real time. RPGs also aren't limited by their special effects budgets, either. (Vast magical worlds populated by diverse creatures, both wondrous and monstrous? I'll have that on the set tomorrow, Mr. Producer. ;-)

How do you win?
RPGs are cooperative play as opposed to the competitive play that you find in most other games. If you play a board game like Monopoly, all the players are competing against each other to see who will win at the end of the game. In an RPG, the players cooperate with one another to accomplish their goals. Moreover, even though the GM will often play NPC opponents of the PCs, the players aren't competing with the GM either. There is no final "winning" in an RPG; RPGs are open-ended, and last as long as people are interested (and have time).
However, there is a very strong sense of accomplishment and reward. Quite often, completing just a single task ("rescue the princess from the ogres") will take several sessions. Assuming you play once a week, that means that it might take a couple months of real time to accomplish a major task. In addition, as your PC completes tasks they gain experience points which are then used to gain levels. This is very much like a video game, where your character becomes more powerful the longer you play. High level PCs are treasured by their players, earned through months (and sometimes years!) of challenging play.

What do we do?
Every session the group extends the current plotline. The typical story told is that of an adventure, which has a distinct beginning, middle, and end. A sequence of related adventures form a campaign, which has many stories woven together over a long period. Campaigns usually take place in a distinctive setting; settings are also called game worlds and occasionally backdrops.
The typical D&D adventure is very simple. The PCs band together and form an adventuring party to meet some threat or accomplish some goal. Usually that goal entails leaving the safety of civilization, journeying to an exotic land, or visiting a strange place. There the party faces peril from ferocious monsters, pernicious traps, mysterious magics, and ingenious puzzles. After overcoming all obstacles, the PCs gain treasure, items of power, and abilities far beyond normal people. Then they return to civilization to enjoy their hard-won spoils... and probably do the same thing next week. ;-)

During the session, the GM describes the current situation, and then the players declare what their characters do. If the action is ordinary ("I go buy a sword") then it normally occurs without incident. But if the action is extraordinary ("I want to jump the 20 foot chasm") or against an opponent ("I try to hit the evil orc with my shiny new sword!") then the outcome is decided by a die roll.

The standard die roll in a d20 game works as follows. You declare your intent to do something and the GM decides the Difficulty Check (or DC) of your task: this is the number "to beat". You roll a d20 (a die with numbers 1 thru 20 on it), add any relevant modifiers (plusses that help your chances and minuses that hurt them), and compare it to the DC. If you get the DC or higher, then you've succeeded! If you roll lower than the DC, then you have failed. Depending on the task, failure might be harmless or carry its own consequences (like falling down that chasm).

In order to give credit to those who deserve it:

http://www.ugcs.caltech.edu/~kel/KDD/playerintro.shtml#index1