Jumping spiders don't have large muscles for jumping like grasshoppers. Instead they have a well developed hydraulic system. By altering the pressure of the body fluid (hemolymph) and releasing this pressure to extend their limbs, the jump is initiated.
The limbs stretch within a few 1/1000 of a second and catapult the spider forwards. The amount of pressure controls the length of the jump. Most jumping spiders can jump several times the length of their bodies.
Before jumping the spider fixes a silky dragline to the ground. If it misses the target or it gets away, the spider can climb back to the point of origin on this safety line. It is suspected that the spider also stabilizes the position of it's body during the jump with this dragline to ensure a precise landing onto the prey.