The rectifier diode is simply a one-way conductor. Applied in an AC circuit it will pass only the positive or negative phase of a sine wave. Rectifier diodes are rated for operating voltage and maximum current. A rectifier bridge is composed of two or four separate diodes. Its rating is the total of all diode ratings on the bridge.
Bridges are useful as they are already internally connected.
Fullwave rectification is where both the positive and negative phases of a sine wave are rectified. Halfwave rectification is where only the positive or the negative phase is being rectified. Halfwave rectification is used where the DC voltage need not be smooth and for economy. The capacitor is used as the primary filtering element of the rectified voltage and as an added bonus serves as a tank to help maintain a constant voltage level as the load demands. I have my own formula make them as large as you can fit them in capacitance wise and double the voltage of the rectified output.