Broadly speaking, a thermite is simply a mixture of a metal oxide and a more-reactive metal. When ignited, the oxide gives up oxygen, which then combines with the other metal, making a different oxide.
The thermite mixture that most people are familiar with is Iron Oxide (aka "rust") and Aluminum. This mixture is extremely hard to light, but once it does get going, it produces a slag of aluminum oxide floating atop a blob of molten iron as its reaction products. Railroads used to use this thermite mixture to weld train tracks together.