A skydiver's parachute increases the skydiver's velocity by increasing his cross sectional area. Air resistance is affected by two factors: how fast the skydiver falls and his cross sectional area.
As the skydiver falls, he falls faster and faster, and his air resistance increases until he reaches terminal velocity, which is the fastest speed that he will fall. But the skydiver's air resistance can also be affected by another factor: his position with respect to the gravity. This is why a skydiver usually dives spread eagle; if he were to dive feet first, he would not have as much air resistance and would reach an even faster terminal velocity.
However, the cross-sectional velocity of the skydiver alone is not enough to slow him down a sufficient amount so that he can land safely. This is where the parachute comes in. When the parachute is opened, it covers a large area as the wind opens it up. This greatly increases the air resistance, and is the reason why there is usually a snap up when the parachute opens and the cross sectional area of the combined skydiver/parachute suddenly increases greatly. This ultimately slows the terminal velocity of the skydiver down to the point where he stops moving.