Schist is a metamorphic rock. That is, it didn't always look like schist. Once it was a happy little basalt or shale or slate, and then it got heated and squashed. The result is a rock that breaks into thin layers. The layered structure is termed, believe it or not, the schistosity (now there's a word you don't get to use every day). If you'd like another fancy term, schist is also said to be strongly foliated.
Schists are composed of relatively coarse mineral grains, with a grain size larger than a millimeter. Similar rocks with a smaller grain size are called phyllites. A useful rule of thumb is that if a rock is nondescript and you can pick flakes of mica from it, it's probably a schist.
For more detail, you'll find schists named after the minerals that predominate in their makeup. Thus, there are mica schist, hornblende schist, talc schist, garnet-muscovite schist and so on. Thus, schist refers more to the schistosity of the rock than anything about its composition, though most schists include large amounts of quartz.
Because it's been all squashed, schist is a very hard rock that doesn't squash much more. That helps explain why Manhattan Island in New York is home to so many skyscrapers: it's largely underlain by schist, which makes a good base for all those heavy buildings. Another place you will find schist is at the very bottom of the Grand Canyon, where there is a stratum called the Vishnu Schist. This particular rock layer is a schist with large quantities of mica that formed 1.7 billion years ago or so.