On any kind of aeronautical chart you'll see many VORs (or VORTACs), which have been the main enroute navaid for the last forty or fifty years. Think of them as a wheel laid on the ground with 360 spokes, which are the 360 radials, one degree apart, with the zero-degree radial pointed at magnetic north. The chart shows the radio frequency for each VOR; when an aircraft with the necessary equipment tunes to a VOR the pilot sees that he is on the 90-degree radial of that VOR (if he is directly magnetic-east of it) or on the 315-degree radial (if he is NW of it) and so on.
So shortly after a flight from New York to Boston takes off from La Guardia Airport, the controller will probably tell the pilot something like "Delta eighteen twenty eight turn left zero four zero, intercept the La Guardia zero five five, resume own navigation." In other words, "Point the aircraft 40 degrees right of magnetic north until you reach the 055-degree radial of the La Guardia VOR, then turn right and fly along it according to your flight-planned route." (As it happens, aircraft from LGA to BOS use the VOR at the airport, but that's often not the case; aircraft leaving SFO and OAK to fly to Los Angeles are invariably told to intercept the 135-degree radial of the Point Reyes VOR, which is many miles away to the NW.)
Airways over land (in developed countries, anyway) mostly run from one VOR to the next, which means they are defined by which radial they use at each VOR.