It depends on what freqency and power level the RF waves are to some extent. The basic blocks are:
An antenna to convert the radio waves to electrical signals, or perhaps a direct tap off the transmitter.
A tuned circuit or filter to select the wavelength
A detector to convert RF signal to a DC signal suitable to operate a relay.
A relay driver which amplifies and buffers the weak DC signal to operate a relay.
The relay circuit itself which needs to have contacts that handle the intended loads voltage and current. This in turn determines how much drive is required for the relay coil.
A power supply (say a 12V DC plug pack) to operate the relay and its driver circuit. In some cases a 6 or 9V battery may be sufficient.
The most basic device is something like the crystal set of old. The output is rectified RF, which is filtered to get the DC that is used to trigger a relay circuit. This will work for AM or FM and most digital kinds of system like mobile phones, though it gets tricky for mobile phones because of the high frequency and weak signals. If it is SSB there is no continuous carrier so you need to rectify the audio output as well. The diode needs to be fast enough to work at the frequency of interest. The sensitivity of this arrangement is not high - it will work only with a nearby or high powered transmitter or a large antenna. It typically needs a few milliwatts from the antenna or pickup.
If it is just AM or FM broadcast a cheap radio is one way to get the trigger signal, that requires little knowledge and is probably cheaper in the long run. If there is audio continuously you can just rectify (detect) the signal at the loudspeaker. If there is no audio sometimes, perhaps you can break into the radio circuit and pick off a DC signal from the AGC line or detector. Look for the detector diode.
The weak DC trigger signal can be used to operate a relay using an op amp, a 555 timer, and/or a transistor to increase the drive to operate the relay.
The links below could be useful.
Crystal radio circuits:
Just replace the headpones with a resistor, perhaps 22Kohms. Use a small RF schottky diode for better results especially at higher frequencies. Power diodes won't work.
The relay driver (second link) converts a weak and high resistance DC signal to one able to operate a relay.