Hypergolic rocket fuels spontaneously ignite when their two components come into contact with each other.
One important engineering advantage is that they prevent a rocket motor from igniting explosively, and exploding. This is called a "hard start " and is one of the most destructive things that a rocket can do. However, hypergolic fuels are highly toxic and sometimes unstable, so a very high quality igniter, like a small rocket engine, is often used as an alternative.
Another important advantage is that the igniter cannot fail, so hypergolic fuels were the solution when the rocket must really start as is the case with intercontinental ballistic missiles. However because of difficulties in storing fuel, the trend in ICBMs has been to move toward solid fuel boosters.
Some common hypergolic fuel combinations are: