StagingStaging is the use of multiple independent rockets to reduce the total amount of mass that needs to be accelerated. As the rockets, known as stages, run out of fuel, they are discarded.
With this system the final mass of the rocket is lower than it would otherwise be, as empty fuel tanks are thrown away. On the downside, staging requires you to loft engines which are not being used until later, as well as making the entire rocket more complex and harder to build. Nevertheless the savings are so great that every rocket that launches payloads into orbit uses staging.
Most rockets use linear staging, in which a number of rockets are stacked on top of each other and fire one after the other. An example of such a rocket is the Saturn V. In order to increase the effciency of the staging, the "upper stages" were fueled by hydrogen, meaning there was much less mass to lift than had they used kerosene.
Soviet designs have tended to favour overlapping stages, where the "next" stage fires before the previous one is disconnected. This is why many Soviet rockets have "cages" between their stages, to allow the rocket exhaust from the upper stage to blow out and around the stage below it. The usefulness of this technique is questionable unless you have engines that take some time to get up and running, which is the case for the Soviet designs that used many small engines.
Many designs have been made to use parallel staging in which a number of stages fire at the same time. Early rockets in both the US and USSR have used such designs. Today many rockets that formerly did not use this technique have started to via the use of "strap on" solid rocket boosters to increase the delivered load. A good example of this is the original Thor (a development of the V-2) which evolved into the Thrust Augmented Thor, and finally to the Delta. Another example is the Space Shuttle which fires its solid rockets in parallel with its main engines.
Several attempts have been made to build completely parallel stages, in which an economy of scale could be achieved by using a large number of identical stages strapped together into a bundle. The most complete was the OTRAG project, which failed for political reasons.
In more recent times the usefulness of the technique has come into question. As the costs of space launches appear to be almost entirely the operational costs of the people involved (as opposed to fuel or other costs), reducing these costs seems like the best way to lower the costs. Since staging is expensive in terms of manpower, a new movement has concentrated on SSTO designs that have no stages.