Rather than being a specific formula, as is often misunderstood, thermite is a term for a group of chemical mixtures that, when ignited by a sufficiently hot source, react to give extremely high temperatures. Thermites and variants on them have a variety of civilian and military applications, from specialized welding, to incendiary formulations, to destruction charges used to make equipment useless if captured.
The basic mixture in a thermite is a powdered metal, most often aluminium powder, and a metal oxide, most often ferric oxide. With a pure thermite, the heat is generated not by combustion, but by an reduction-oxidation reaction. There is no flame or flame propagation. The reaction, using iron oxide and aluminium, was patented by Goldschmidt in 1895, and is characterized by: 
- Minimal gaseous products
- High temperature, typically 2000 to 3000 degrees Celsius; specialized thermites can be prepared for a specific desired reaction temperature
- Production of a slug of molten metal, which can transfer heat efficiently for burning, cutting or welding
If metal oxides of high specific gravity, such as the oxides of tungsten, molybdenum or lead are used, the thermite reaction is especially active, although the mixture can be especially expensive. Nevertheless, these characteristics can be especially important in specialized applications, such as flares used as decoys in electronic warfare.
While the igniter is not part of the mixture, it is critical to thermite applications. The igniter needs to provide a high temperature; magnesium metal ribbon, which can be lit with a match, is a common igniter for ad hoc thermite work.
Thermates add pyrotechnic materials, such as nitrates (most often barium nitrate). They are easier to ignite. Combustion reactions do take place with these substances, and gases are produced. The gases project the liquid metal slug and give increased penetration, desired in incendiary and other specific applications.  U.S. Army "thermite" grenades actually contain a thermate, which, in the TH3 grenade, is a mixture of thermite 68.7%, barium nitrate 29.0%, sulfur 2.0% and binder 0.3%. 
Thermites and thermates are used in specialized welding and metal-cutting applications. The reacting mixture, and the molten metal it produces, often needs to be contained by ceramics, or it may cut away its own support and fall away from the work. It normally, therefore, needs to react on a horizontal surface. This requirement does throw doubt on some conspiracy theories associated with the 9-11 attack in New York, which suggest that preplanted thermite was used to cut vertical supports in the World Trade Center.
One application, which lends itself to the horizontal placement, is welding railroad rails. A relatively small amount of thermite can replace cumbersome gas welding tanks or the need for electric power for arc welding. This type of welding is used as a spectacular demonstration in high school chemistry, although it generally is too dangerous for inexperienced students to perform.
It was first used as a military incendiary in the First World War, dropped from German zeppelins. The German formulation was of magnesium powder and magnesium oxide; the French introduced the aluminium-iron oxide. Allied troops also introduced destruction grenades that raiding units used to destroy German equipment.
The great heat of thermite reactions obviously can cause burns, but there may be other means by which thermite constituents do damage. Metal particles may burn even more deeply than the flame. Those metals that are reactive, such as magnesium, can react chemically to produce alkali, which in turn produces chemical burns. The temperature of the reaction is sufficient to produce light that can burn the retina. 
The thermite reaction is an example of an reduction-oxidation reaction, in which one compound becomes reduced while another compound becomes oxidized. In the prototypical thermite reaction shown below, aluminium, which is a good reducing agent, reduces the ferric oxide (Fe2O3, rust) to elemental iron (Fe) while the aluminium powder (Al) itself becomes oxidized to aluminium oxide (Al2O3). A large amount of heat is released in this nearly unstoppable reaction, about 850 kJ/mol), thus making it very dangerous to attempt.
- Eq. 1: Fe2O3 + 2Al 2Fe + Al2O3
The thermite reaction is most often started by adding compounds that are easier to ignite. Magnesium ribbon can be placed on top and ignited by a flame source, or a few drops of glycerin can be added to potassium permanganate (KMnO4). Barium compounds such as barium oxide (BaO2) or barium nitrate (BaNO3) might also be added to the mixture.
- K. Kosanke, B. J. Kosanke, I. von Maltitz, B. Sturman, T. Shimizu, M. A. Wilson, N. Kubota, C. Jennings-White, D. Chapman, Pyrotechnic Chemistry, Journal of Pyrotechnics Chapter 8, page 18
- Kosanke et al., chapter 8, pp. 20-21
- E. Song, U.S. Patent 6766744: Incendiary device
- Anthony L. Feliu (1 January 2001), "(Abstract) Report: Thermite Welding Gets High School Chemistry Class on Track", J. Chem. Educ. 78 (1): 15, DOI:10.1021/ed078p15
- Military Explosives, U.S. Department of the Army, September 1984, TM 9-1300-214, pp. 2-10 to 2-12
- Richard Adam Koch and Robin A C Marshall (6 August 2008), "CBRNE - Incendiary Agents, Magnesium and Thermite", Medscape