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Usually Full and half bridges refer to rectifying AC to DC. A half bridge has one diode so that the only the positive (or bottom) part of the sine wave get through.
A full wave uses 4 diodes but instead of just clipping off half the wave it reverses the polarity of half the wave, thereby increasing the efficiency and doubling the frequency.
After the bridge you would usually put a filter and a shunt cap to smooth the wave into DC and to remove any ripples.
An H bridge uses switches/transistors to alter the flow of current so as to reverse the polarity applied to the motor.