A radar altimeter is a device used by aircraft to determine their altitude relative to terrain - that is, their height above the ground. They are standard equipment on many aircraft equipped for precision instrument approaches, especially CAT I, II or III.
The 'standard' altimeter used in aviation is a barometric device, which relies on the ambient air pressure outside the aircraft to determine the height above ground. While this is a handy and simple measurement to make mechanically, it is complicated by the fact that ambient pressure is not solely related to height, but also to weather conditions. In addition, the barometric altimeter does not return ones height above the ground, but above sea level. Since ground level is most often higher than that, it can be dangerous to rely on this number unless you also happen to know exactly how elevated the terrain below you is.
The radar altimeter (also known as the radio altimeter or just RA in some aviation circles) was invented in 1924. It was first placed in aircraft in 1938, packaged by Bell Labs. This device operates in one of several modes, depending on which type it is but at base, it determines the distance to the ground by bouncing a radio signal off the surface of the ground below and measuring the distance to that surface (hence, radar).
The RA is most useful in instrument approach procedures, where the height above the actual surface is critical. It is typically used at low altitude, as its range is limited - under 1000 meters or so. It will report to the pilot and to the autopilot, autothrottle and/or flight management system depending on how the aircraft is configured. In many cases, where an aircraft is certified for autoland, the RA is used by the autopilot to determine when to flare the aircraft for landing as well as to ensure it is on profile during approach.
There are several varieties of RA depending on their operating mode. The simplest uses what is called Pulse-Limited Altimetry which uses basic time delay information from the radar pulse and reflected signal. Another, Frequency-modulated continuous-wave, uses the frequency phase offset between the pulse and the return signal to determine the range to the ground.
Other systems can be used to provide radio altimetry. On some airliners, the weather radar can be configured to look straight down and perform the same type of measurement with the higher power budget and hence range of the system. Military aircraft sometimes have terrain following radar which is a continuously updated RA used for navigation, which may also be hooked up to the control systems to provide terrain avoidance system to automatically climb the aircraft if it detects imminent collision.