An exoatmospheric kill vehicle (EKV) is an orbital weapons platform whose primary purpose is the destruction of satellites, ICBM warheads, and other man-made space-faring objects.
In principle, a payload launch vehicle (PLV) takes off from Earth, and when it reaches space it's payload, the EKV, is launched. This has it's own sensors which it uses to steer itself towards the target using thrusters. There are no explosives involved - the target is obliterated by the kinetic energy in the (hopefully) resulting collision. This is the 'hit-to-kill' principle, which has been likened to 'hitting a bullet with a bullet'.
Conceived during Star Wars, the first working concept was demonstrated in 1984 after six years of work and numerous test flights. It was effective and extremely high-tech for it's time, but also heavy and expensive.
The current scheme uses modified Minuteman II ICBMs (minus warheads, plus modern electronics) as the launch vehicle, courtesy of Lockheed Martin. Boeing and Raytheon are developing competing EKV payloads. Both use infrared sensors to lock on to their target.
Testing with the current system has met with mixed success, with (at time of writing) two out of three tests failing. The project is classified "Top Secret". and the whole deal is overseen by the US Army Space and Strategic Defence Command.
P.S. There is some variation on exoatmospheric or exo-atmospheric. The first turned up more hits on Google.