How does a bridge rectifier work exactly?
A simple bridge rectifier circuit is as shown :



During the positive half cycle, diodes D2 and D3 are forward biased and D1 and D4 are reverse biased. Hence the direction of flow of current is D2 -> Load Res -> D3.

During the negative half cycle, the diodes D4 and D1 are forward biased while D2 and D3 are reverse biased. Hence the direction of flow of current is D4 -> Load Res -> D1.

An important point is that the direction of current is the same ie from right end to left end through the resistor, and hence the polarity of voltage obtained does not change.

You can find additional details here :

http://www.electronics-tutorials.ws/diode/diode_6.html