Drum triggers come in many forms but fall into a two main categories.
The first could be considered the acoustic type, and it is best explained through example: Heavy bands often require a very specific bass drum sound--one that might not be acoustically possible in a certain venue or studio. To achieve a consistent bass drum sound no matter where that band performs, the band may use a drum trigger. A microphone is placed inside the bass drum and wired to a processor that contains the desired drum sound. Whenever that drum is played, the processor produces that sound.
The second could be considered the synthesized type; instead of the sound being triggered by a drummer playing a traditional kit, it is initiated through the push of a button or the attack of a drumstick on a pad, for example.
As complicated as drum triggers may sound, they are essentially very simple and require only two elements: a sound to be triggered and something to produce the electrical impulse needed to trigger it.
Drum triggers don't necessarily have to initiate drum sounds. You could trigger a snare drum to initiate a prerecorded piano sequence, you can use a bass drum to initiate a guitar part or you could use a floor tom to initiate the sound of a duck quacking. The options are limited only to the sounds available to the recording or sound engineer.
Because drum triggers make use of modern music technology, some people wrongly associate them with industrial, electronic or heavy bands that rely on synthesizers as a staple of their sound. While those types of bands do use drum triggers, so do bands in every other genre. Drum triggers are not used to produce a certain sound or effect; they are used only for consistency.
Perhaps the most famous user of drum triggers is Rick Allen, the drummer for the English hard rock band Def Leppard. Allen lost his left arm in an automobile accident after the recording of the band's multiplatinum "Pyromania" album. For the recording of the band's follow-up, "Hysteria," Allen used a custom-made drum kit that allowed him to play his left-hand parts through the use of drum triggers he operated with his feet.