I'm a frequent flyer on several airlines. Dave's correct -- you sign up on the airline's Web site, or open a credit card affiliated with the airline. You can usually link to the credit card application directly from the airline's site. You earn miles by flying on the airline (or on one of its partners--see Web site for details), or by using the credit card.
Nearly every airline has a frequent flyer program. Some belong to the same network, so you can earn miles on one and redeem on another. For example, Delta, Continental, Northwest, and Air France are among those that are linked through a program called Sky Miles. (Many restrictions apply, though.)
As LHP said, redeeming miles has become far more difficult. Several airlines now expire those miles at 18 months since last activity, down from 36 months. Many will also restrict the number of freebies at the stated rate, and require you to use double miles. (In other words, if 25,000 miles is needed for a free flight, you can use them only on certain flights OR shell out 50,000 miles to go when you want.)
A secret to keeping those miles from expiring is to have some sort of activity, even if it's not flying. For instance, you can preserve miles on an airline by presenting your frequent flyer number when renting a car, for those rental companies that will accept them.
I redeemed miles for free airline tickets only once -- turned in 100,000 for a free flight for me and my son for our Arizona trip last year. (Yes, that's double miles.) I actually prefer to use my miles for first class upgrades on long flights. Not only does it make for a more comfortable flight, I still accrue some miles as I've paid for a coach ticket.