An ordinary (linear) supply uses a transformer to change the voltage to approximately what is needed. Then circuitry changes this to DC, ensures that it is clean and remains at the proper level (rectification, filtering and regulation). The problem with this design is that line frequency (50/60 Hz) transformers are big, heavy, and expensive.
The key to the operation of a switching supply is operating the transformer at a much higher frequency, most often beyond audible frequencies. At higher frequencies the iron core of the transformer is no longer needed (in fact it would prevent the transformer from working). This makes for a smaller, lighter transformer in the supply.
How does it work?
Line voltage (AC 120 volts) is rectified when it comes in, resulting in a DC voltage approaching 200 volts. This DC voltage is fed into a high power transistor circuit, which is designed to oscillate at the (high) design frequency. The transformer, in addition to being part of the oscillator circuit, does what transformers do, and makes the high frequency, high voltage AC available through its secondary windings. From the secondary, the AC passes through a rectifier, which converts it back to DC (the higher frequency source is now irrelevant because it is DS) filter (to smooth out any remaining ripple), and (voltage) regulator as above in the linear supply.
The result of all this is a supply that is lighter, smaller and potentially more stable than an older linear design.