REV MATCHING is a driving technique for shifting a manual transmission smoothly. One simply memorizes approximately how many RPMs different the various gears are from one another at the same speed, and then matches the engine's revs with the gear they are shifting in to. When upshifting, you just let the revs drop and slip into gear. When downshifting, you tap the throttle (or stomp on it depending on how long it takes to rev up.) This is often referred to as 'blipping' it. This works easily primarily because of the syncromesh gears in your transmission; if your transmission does not have them, this is very, very difficult to do without grinding. Even so, it takes the right touch.
For example, in one car the gears might all be approximately 1,000 RPM apart at the same speed. When downshifting from third to second, for example, you would apply accelerator pressure to raise the engine RPMs by 1,000 (say, from 3,000 to 4,000) and make your shift. You can then use engine braking to reduce your speed.
This practice results in reduced brake wear, less of a lurch when shifting from a higher gear to a lower one, less clutch slippage when shifting from a lower gear to a higher one, and a smoother shift without having to slip the clutch so much.
In auto racing, one often uses the more advanced form of this activity, heel-toeing (Alternately heel-toe braking or heel-toe downshifting) in which one's heel is on the brake pedal, decelerating, while the toe is on the gas pedal, accelerating. This allows the use of both the brakes and the engine to perform braking.