In practice, the half-wave rectifier is used most often in low-power applications because of their major disadvantages being. The output amplitude is less than the input amplitude, there is no output during the negative half cycle so half the power is wasted and the output is pulsed DC resulting in excessive ripple.
A full wave rectifier is a component or set of components that change AC waveforms into DC. It is frequently used as part of a power supply circuit and therefore it is often directly adjacent to a transformer. However, a transformer is used to provide a suitable AC voltage and is not part of the rectifier.