It's a mix of finely ground aluminium (not the oxide), and Iron (III) oxide (i.e. Fe2O3). In fact it only works with the gamma allotrope of the iron (III) oxide.
You need to kickstart the reaction with some serious stuff, for example, lighting magnesium ribbon. Once you kick the reactants over the activation energy, a substitution reaction occurs, with the aluminium taking the oxygen away from the iron, leaving you with molten iron coming out at 3000 kelvin or so, in other words, bloody hot! It's usually enough to melt the reaction container. Hence its name (thermos being the Greek for heat).
Although there are anarchic uses of thermite, there are also practical ones; and this is where it came about. When they lay railroad tracks, the rails come in lengths of a few meters; but this isn't really very useful, so you have to somehow "weld" the rails together. But surface welding is obviously out of the question if you're going to have hundred-ton juggernauts running over the top of it. What to do? Get a mold in the shape of a rail with a small hole on top, insert two rails in either side, put a container of thermite on top, get the reaction started, let the molten iron fall into the gap, wait for it to cool down, remove the mold, bit of sanding and polishing and voila, you have a perfect, solid, all-through weld.