Let's face it, you're going to Disney World to enjoy some rides, right? At least, that's one purpose of your trip. And rides mean lines. Sometimes REALLY long lines, if the ride is very popular. Disney does an amazing job of making the lines to most rides really entertaining and fun (a notable exception is the Speedway Race Cars, where you just stand in a drab line and breathe fumes). But even the most whimsical, creative, and interesting line in the world gets really old if you've been standing in the same spot for the past 15 or 20 minutes.
Enter the FastPass system.
The FastPass system stands in line for you. Basically, it is a slip of paper that gives you a designated window of 1 hour to come back to the ride. When you come back within that window, you go through the FastPass line, which is generally short and sweet (the longest I ever waited was 15 minutes, on the weekend before Christmas). You walk past all the people waiting in the regular 60+ minutes line, and if you're like me, try really hard not to snicker.
Here's an example: The Standby wait time for Splash Mountain on a particular day at 11 AM reads 75 minutes. You walk to the FastPass machine situated next to the ride's entrance, insert your park admission ticket, and the machine gives you back a small slip of paper that says your return time is between 2:20 - 3:20 that afternoon. You spend the next 3 hours doing other fun stuff at the park and having a great lunch, come back at 2:20 and walk straight through.
Or you can spend 75 minutes standing in line in the sweltering FL heat. Some people, for some reason, do this.
The sign above the FastPass area will let you know what window of time the machines are currently dispensing, so if you have a meal reservation, for instance, you won't find out when you get your pass that your window conflicts with your reservation. You can decide if the window works for you, or come back later to see if a later window is available.
Be aware, though -- FastPasses do run out. Depending on the ride and / or time of year, they can run out as soon as 10 AM (1 hour after official park opening). I've seen this happen with Soarin' a lot.