Radio receivers amplifies and tunes the radio signals that is ever present in the air all around us. We cannot hear these signals and if we could, it would be chaos. It would be like being in a large auditorium with everyone speaking at once.
The receiver picks up these signals from the airwaves, and converts them to the sound waves that we can hear. The radio signal that's transmitted into the air contains a carrier wave that's much (thousands of times) higher in frequency that we can hear. A dog can hear frequencies higher than us humans but even they cant hear radio waves. The radio carrier wave is modulated (mixed or Heterodyned) with sound waves that we an hear but they are so weak that we can never hear them without the magic of a radio receiver.
A TRF receiver amplifies an tunes the raw radio signal as present in the air waves. It does this by means of an RF (radio frequency) amplifier. Some receivers will have as many 4 or 5 stages of RF amplification before the carrier signal is stripped away leaving only the audio portion of the signal. The process of removing the carrier signal is done by the detector circuit of a radio receiver. Afterwards the final process is amplifying the audio signal to a level strong enough to drive a speaker.
A super-heterodyne receiver treats the carrier signal a bit different. Instead of amplifying and tuning the raw signal again and again (as does the TRF receiever), a typical super-het may have one or two stages of RF amplification and them mix the raw signal with another signal that's generated within the receiver itself. This is called; heterodyning, converting or mixing. The result of this action generates still another signal; the IF or intermediate frequency. Using this method of amplifying this single radio signal provides several significant improvements over the TRF type circuit. Among other things greater sensitivity and selectivity. The most visible significance is the smaller chassis with fewer components. The IF signal is then treated much in the same way as the TRF receiver, after amplification the detector circuit strips the IF signal away and then the audio circuit does the rest.
There are other significant differences in the two receiver types but hopefully this provides a brief explanation for the non technician.