At its most basic, legal drag racing consists of two cars lining up next to each other on a starting line at a drag strip and on a signal to go, both cars accelerate as hard as they can for a fixed distance, usually 1/4 or 1/8 of a mile. The first one to the finish line wins. Quickness counts, but speed is only an interesting side effect. If a car leaves before the start signal is given, it loses by virtue of a foul. What could be simpler?
If two cars do this sort of thing on the street, which is illegal, legitimate drag racers call this street racing, not drag racing.
As you can imagine, the simplest form of drag racing didn't last too long. Cars aren't all alike, so they were divided into classes based on car weight and modifications to the engine and chassis or the lack thereof. This is class racing. Cars in the same class race each other in round robin fashion until there is a single winner. Class winners in similar classes raced each other in eliminator brackets using some sort of handicap system so that, theoretically, both cars would have an equal chance of reaching the finish line at the same time. In the early days, this was done by starting the cars at different points on the drag strip. Later one car was given a head start by electronic starting devices. This allowed for very precise handicaps based on the class's national record of the elapsed time required to get down the drag strip. Class racing is still done at mainstream national or divisional races, but local drag racing has taken a different turn.
Sometime in the late '60s and early '70s, money began to be a huge factor. The old saying that you can't beat cubic inches (referring to the size of the engine) became you can't beat cubic money. The cost of being truly competitive in drag racing was getting out of hand for the average person. Enter bracket racing.
Since we could now give very precise starting advantages to one car or the other, why not let the driver pick the elapsed time for the vehicle? Less money, slower car. More money, faster car. But both have the same chance of reaching the finish line at the same time. But what was to keep a driver from lying about how quick the car could negotiate the drag strip? The biggest liar would be the winner. Enter the break out rule. If a car goes quicker than the prediction, it "breaks out" and loses. If both cars break out, the one which breaks out the least wins. Cars are generally grouped into large brackets primarily based on elapsed time. This bracket racing is the predominant form of racing at local drag strips.
A variation of bracket racing is having a preassigned elapsed time index, such as 7.90 seconds, where every car has the same predicted elapsed time.
Of course there are lots of subtleties to all this, but this is the basic idea. If you want to learn more about the bracket racing end of things try this link:
Nostalgia racing draws from both "heads up" racing, where the quickest car wins; bracket racing, where every car has an equal chance to win no matter how fast or slow it is; and index racing, where everyone has the same predicted time. I'll try to keep these straight when reporting on nostalgia races, but you may safely assume that anything ending in "Fuel," like Top Fuel and Junior Fuel, are heads up racing.