A style of start used in track and field(as well as other types of races, such as speed skating) when a race is run that incorporates one or both of the turns in the track. Basically, it sets the runners in the outer lanes a bit ahead (usually a few meters per lane) of the first lane, counting out from the inside.
A staggered start is usually used in races where staying in the lane is required, such as the 400 and 200, although it is sometimes used in longer races, although a cutoff is used in that case, and runners can cut in the inside lane after a designated distance.
If a cutoff is used, there are a number of turns specified, such as a one-turn or two-turn cutoff. This means that after completing one turn or two turns, respectively, the runners my cut in to the inside lane. This is often used in 800s, although it can be used in 1600s and 1500s, if the field is large enough to prevent a clean waterfall or barrel.
In a staggered start, the highest ranked runners are given the middle lanes. This is because the inside has a rather sharp turn, which isn't very much desired by sprinters. The outer lanes also have a slight disadvantage, due to their false lead. Starting off ahead of the field, they can't see very many other runners, so they have a hard time judging speed. In the 150 dash, the debacle between Michael Johnson and Donovan Bailey, there was much complaint from both of the competitors about the construction of the track. This was because it was only two lanes, but they chose to construct them as if they were the inside two lanes on a 400 meter track.
Staggered starts do allow for very technically fair races, because each runner is running the exact same distance. However, it does prevent the runner from knowing exactly how far ahead or behind they are until the final stretch.