I'm a British person living in Japan and I sometimes hear North American friends speaking about having something called air miles. Apparently a sort of reward scheme where flights are cheaper if you fly a certain amount on an airline. I haven't the foggiest idea how these things work. I get the feeling I might have missed out on some free money when I flew home last time. I've got a planned trip home coming up and it would be nice if I could start to get this airmiles stuff. Can anyone give an idiots guide to how on earth to 'do' airmiles?
There is actually a program called Air Miles which is active in Canada and other places but no longer in use in the USA. This is a reward program where each time you spend a certain amount at participating locations, you get points. These points can be redeemed for merchandise or travel.
There are similar programs from other vendors, usually banks which issue credit cards. These work the same way in that you accumulate points for spending and get a discount up to the value of the fair when redeeming.
Now what you described is different. It is a Frequent Flier program as @neubert answered. In this case, you get points for flying with a certain airline or ones of its partners, which means another airline in the same alliance. You eventually redeem a certain number of points based on an award chart by booking with the airline issuing the points. The availability of reward seats is subject to limitations and availability which is often less than the number of unused seat on the flight.
The key difference between these types of programs is that the former are value based and the latter are based on regions and dates. This means that you can get something of very different value with Frequent Flier points for a set amount of points. With some effort you can extract very good value of Frequent Flier miles, while points issued by credit-cards and similar companies have a fixed value.
They're more typically called frequent flier miles and you get 1 frequent flier mile for every mile traveled on the airline. So if your destination is 1,000mi away you'll get 1,000 frequent flier miles.
You generally have to get a frequent flier mile number from the airline to accumulate them. Any miles acquired before you had that number are usually just lost.