FASTPASS is a virtual line. Rather than standing there wasting time waiting in line, you can spend time doing ANYTHING else, and then return to the ride/attraction at the later specified time.
Not all rides/attractions at WDW use FASTPASS. The ones that do will have FP machines outside of them, usually 5 or 6 machines or so. Your park ticket (in most cases) is like a credit card. You put your ticket in, it goes inside the machine, and a moment later the card is returned to you through that same slot, and at an opening near the bottom of the machine there appears a piece of paper called a FASTPASS, about the size of a movie ticket.
This FP has 2 important bits of information on it (besides the name of the ride and the date). One bit of information is called the return window. It will be a one-hour span of time. This is the span of time in which you are permitted to use the FP. Yes, many people have had the experience of being able to use their FP AFTER the return window, but a few people have been turned away and told the FP expired. Use this information as you choose. One thing that seems without controversy is that you won't be able to use your FP BEFORE the return window.
The other important bit of information on the FP tells you when you can obtain another FP. The DIsney computer will not let you go around the park collecting FPs all day. The FP itself tells you when you can get another. Often, this time period starts at the beginning of your FP return window, but it could possibly be earlier. It's often wise to get another FP as soon as you're able BEFORE going on the ride/attraction for which you got your first FP. That way, some of your waiting time for the second ride is spent riding the first ride.
As someone else mentioned, each person in your party must have their ticket go through the FP machine to obtain a FP. In other words, one ticket = one FP. You can't go to the machine and put your own ticket through 4 times, it will only return one FP to you. BUT, each person does not need to be present at the machine. If everyone is sitting and taking a breather, for example, you can walk over with everyone's ticket and feed each one through the machine. Also, FPs don't have your name on them, so you don't need to keep track of who's ticket got which FP.
When you look at the entrance area of a ride/attraction, you'll see the regular line area, and also a FP return area. When you are about to redeem your FP, go directly to the FP return line. Typically, there is a CM right there to check to see that you have a FP for each person about to go through (again, one person can hold all of them). You go through the FP line area, and just before entering the ride/attraction you'll be met by a second CM, the one who will take your FPs. There's nothing electronic or computerized about this transaction -- in other words, Disney doesn't know whether or when you actually USED a FP.
Also, before you decide to actually get the FP in the first place, look at the sign telling you the current FP return time (that is, it tells you what time you can return if you get a FP right now). It may be later than you want, and if that's the case you just wasted the opportunity to get another FP quickly. I did this once and it messed up part of my day -- the return time was too late in the day, I knew we wouldn't still be in the park, but according to the info on the bottom of the FP I couldn't get another one for another 2 hours, so I was stuck with useless FPs and the inability to get more until too late.