A type of jet engine where all the air passing through the engine is used to burn fuel and drive a turbine as well as generating thrust.
The turbojet is one of the simplest types of jet-engine, and the first mass-produced type thereof (motorjet-propelled aircraft date back all the way to 1910, but remained essentially one-offs); the first turbojet-powered aircraft flew during World War II, although the first, experimental, turbojets ran during the mid-1930s.
The air entering the intake of a turbojet passes through a series of compressors (very early turbojets used centrifugal compressors; later types generally used axial compressors, which are slimmer and produce less drag), which compress and heat the air before it enters the engine's combustion chamber, where fuel is sprayed into the airstream and burned. The superheated air and combustion products then expand through a turbine, which is mounted on a shaft, which, in turn, powers the engine's compressors. The engine exhaust, still very hot and highly-pressurised, then blows out the tailpipe, producing thrust.
As turbojets produce thrust only from the high-speed exhaust from the engine core via propelling nozzles (unlike a turbofan, which produces part to most of its thrust from fan-blown bypass air), they consume far more fuel than turbofans in the slow-flight regime, but are economical in the high-speed regime. This fast, hot exhaust makes turbojets many times louder than turbofans; some turbojet-powered aircraft were fitted with improvised "hushkits" to try and muffle their noise, but they were still much harder on the ears than even a modest turbofan, and turbojets' extreme noise levels and extreme fuel consumption meant that airliner manufacturers replaced their turbojet-powered models with ones propelled by turbofans as soon as the latter became available.
On the other hand, a turbojet's hot and fast exhaust is advantageous for very-high-speed flight, and, as such, all early military jets used turbojets (although even fast fighters have since switched to low-bypass turbofans, which have an acceptably fast exhaust - especially since today's fighters are no longer expected to fight at anything much above mach 0.8 - and burn far less fuel); turbojets are the best type of jet engine at speeds between approximately mach 2 and mach 3 (below mach 2, turbofans are superior choices - low-bypass turbofans between mach 1 and mach 2, and high-bypass at subsonic speeds - while, above about mach 3, the ramjet takes the cake).
For more information, see Wikipedia.