From Young Composers
Any combination of drums and non-pitched percussion can feasibly be called a drumkit, but traditionally a basic kit includes a snare, a hi-hat, and a kick drum. Most kits also include a floor-tom, any number of tom-toms, and any number of cymbals. Also, a lot of kits include additional kicks, cowbell(s), roto-toms, and anything else the drummer feels the need to set up to be hit with a drumstick. To understand how to play or write for a drumkit, you have to understand each individual piece. Additional names for this setup are drum set, trapset, or kit.
The Pieces
Drumstick
Almost all drumkits are played with drumsticks, which are basically sticks of wood with a head on the end of it that are used to strike the various components of the kit. Sometimes the head is just a part of the wood, and sometimes it's plastic. If a proper drumstick is not available, a drummer may opt to use anything that he can strike the components of the kit with. It's a good idea to take into account what sort of damage an object could inflict on the kit before using it as an impromptu/makeshift drumstick.
Hi-Hat
The Hi-Hat is two cymbals placed on on top of each other so that the leading edges face each other. The top cymbal is attached to a rod that is attached to a foot pedal. This pedal is used to move the top cymbal up and down. This allows a hi-hat to have many different possible sounds, as the cymbals can be struck independently, held tightly together, held loosely together, held slightly apart so that they still hit each other, or transitioning from one position to another. The hi-hat is most commonly used to keep a steady beat.
Kick Drum
The kick drum is a usually the center piece of the kit. It is a bass drum on the floor directly in front of the drummer, and is struck with a felt hammer which is operated with a foot pedal. It is possible to have two kick drums, one for each foot. A similiar method is to have two pedals that operate two hammers right next to each other which strike one drum. Both of these methods are called a "double-kick". It is also possible, but rare, to see a quadruple kick. On a quad-kick each pedal controls two different sets of two hammers which are staggered so that they hit the drum at different times. This can allow for what is essentially a bass drumroll.
Snare
The snare is the primary drum outside of the kick drum. It has a unique sound produced by strands of metal stretched across the bottom head. It is normally used in conjunction with the kick drum and the hi-hat or a cymbal to produce the main rhythm.
Crash and Splash Cymbals
Crashes and Splashes are cymbals which are designed to be very loud and pronounced. They most often are used sparingly for dramatic effect.
Ride Cymbals
Rides are cymbals designed to be more mellow, and are commonly used to keep a steady beat. In general, striking a ride hard can also produce a similiar effect as a crash or splash, but is usually not suggested as they are not designed to be treated as roughly as crashes and splashes.
Tom-Toms and Floor Toms
Toms are generally the most melodic part of the kit. A kit can have any number of toms, or none at all. When there are more than one, they are usually lined up around the bass drum and tuned in descending order. Toms are commonly used for "rolls" where the drummer will usually start at the highest tom and play a short drumroll on each one, progressing down to the lowest. Floor toms are the lowest pitched toms.
Additional Notes
A kit can also include anything else the drummer deems fit. When writing for a drumkit the most important thing to remember is that the drummer can only strike as many components as he has arms and feet. A common mistake is to write a part with extensive double-kick and a transitioning hi-hat. When a drummer is using both feet for his double-kick, it is impossible for him to also use the foot pedal to operate his hi-hat. He must either take his foot off of one of his kick pedals to operate his hi-hat, or he must simply let the hi-hat stay in one spot. Another common mistake is to write a part with a fast roll on the toms that require both hands, while also hitting the hi-hat or cymbals.
As with any instrument, some drummers are faster than others. Some drummers can only play a single kick, while some drummers can't play complicated hi-hat tricks. It is important to know the skill level of the person you are writing for.
See Also
Percussion Staves in Finale