An invertor converts low voltage DC to high voltage AC. It consists of a chopper device which opens and closes a switching transistor successively. This produces a pulsating DC. Then a transformer steps up the current to higher voltage. The output is closer to a square wave than a good sine wave. More expensive chopper will produce a wave form closer to sine wave.
To convert AC to DC, one uses a rectifier. This device uses one or more diode to force/convert current to flow in one direction. The result is a pulsating DC current. The ripples generated can be smoothed out to approach a pure DC form by using filter. The rectifer works by trying to maintain the DC voltage at its peak value.
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