VOR/DME refers to combined radio navigation station for aircraft, which consists of two radio beacons, placed together, a VHF omnidirectional range (VOR) and distance measuring equipment (DME). VOR produces an angle between the station and the receiver in the aircraft, while DME does the same for range. Together, they provide the two measurements needed to produce a navigational "fix" using a chart.
The VOR system was first introduced in the 1950s, but became much more practical with the introduction of low-cost solid state receivers in the 1960s. DME was a modification of WWII-era navigation systems like Gee-H, and began development in 1946. Like VOR, it only became practical with the introduction of solid state receivers during the 1960s.
During the mid-1960s, ICAO began the process of introducing a standardized radio navigation system for medium-area coverage on the order of a few hundred kilometres. This system would replaced the older Low-frequency radio range and similar systems used to navigate over national ranges. A number of proposals were submitted, including ones based solely on angle measurements like VOR, solely on distance measures like DME, combinations, or systems that output a location directly, like Decca Navigator and Loran-C.
VOR/DME eventually won the standardization effort, due to a number of factors. One was that the direct measurement systems like Loran were generally much more expensive to implement (and would be into the 1980s) while Decca had problems with static interference from lightening strikes. The choice of VOR/DME as a hybrid was due largely to it being easier to measure and then plot on a map. With VOR/DME, measurement from a single station reveals an angle and range, which can be easily drawn on a chart. Using a system based on two-angles, as an example, requires two measurements at different frequencies (or using two radios) and then the angles plotted from both on a single chart, which may be difficult in a cramped cockpit.