The term for the ability to 'tune in' to the required signal, and reject others, is 'selectivity'.
In simple terms, an LC (inductor-capacitor) circuit is made. These circuits have a resonant frequency, that is, a frequency at which they oscillate strongly. Just like a tuning fork oscillates at a particular frequency when struck. In the LC circuit's case, it is the current oscillating backwards and forwards through it. The LC circuit is adjusted to oscillate at the wanted frequency; this is done by turning the tuning control on an old-style radio and can adjust either the variable capacitor or, less usually, the variable inductor.
The LC circuit is connected to the antenna, which responds to a wide band of frequencies even though it might itself be 'tuned' to a certain range.
The LC circuit oscillates most strongly at its resonant frequency, and less strongly the further away from this. So, the antenna's signal will have the required frequency amplified, and others diminished, in the output signal. The output of this goes to whatever the next stage is (for instance an RF amplifier; the LC might be built as part of the RF amp). So, we now have a signal that is stronger in the wanted frequency range, and weaker in the others, getting weaker the further away we get from the tuned frequency. Just like with a tuning fork where you cannot even hear anything but the designed frequency.
The earliest receivers, crystal radios, used a single LC circuit, connected through a diode (the 'detector', actually just a rectifier that removed one half of the modulation envelope) to headphones. Note that a wide range of frequencies is still present even after the LC circuit, but the unwanted frequencies (away from the LC's tuned frequency) are very much weaker than the resonant frequency. The headphones themselves (and ever one's ears) act as a low-pass filter so that what we hear is just the audio modulation on the signal.
A single LC circuit is not very selective though, and as stations get closer together, it is not good enough to separate them.
In a very old design of receiver, the Tuned Radio Frequency method would have several of these LC/Amp stages to further refine the signal in the wanted frequency at each stage. However, each stage needs retuning when a new frequency is wanted. That's not convenient, so other methods are used now after a single selectivity stage. Some such methods are Direct Conversion and Super-heterodyne. In any case though, the same principles of resonance are applied in several stages to continually improve selectivity.