A rectifier is a circuit which converts the Alternating Current (AC) input power into a Direct Current (DC) output power. The input power supply may be either a single-phase or a multi-phase supply with the simplest of all the rectifier circuits being that of the Half Wave Rectifier. The power diode in a half wave rectifier circuit passes just one half of each complete sine wave of the AC supply in order to convert it into a DC supply. Then this type of circuit is called a "half-wave" rectifier because it passes only half of the incoming AC power supply.

During each "positive" half cycle of the AC sine wave, the diode is forward biased as the anode is positive with respect to the cathode resulting in current flowing through the diode. Since the DC load is resistive (resistor, R), the current flowing in the load resistor is therefore proportional to the voltage (Ohm´s Law), and the voltage across the load resistor will therefore be the same as the supply voltage, Vs (minus Vf), that is the "DC" voltage across the load is sinusoidal for the first half cycle only so Vout = Vs.

During each "negative" half cycle of the AC sine wave, the diode is reverse biased as the anode is negative with respect to the cathode therefore, No current flows through the diode or circuit. Then in the negative half cycle of the supply, no current flows in the load resistor as no voltage appears across it so Vout = 0.

Very often when rectifying an alternating voltage we wish to produce a "steady" and continuous DC voltage free from any voltage variations or ripple. One way of doing this is to connect a large value Capacitor across the output voltage terminals in parallel with the load resistor as shown below. This type of capacitor is known commonly as a "Reservoir" or Smoothing Capacitor.

When rectification is used to provide a direct voltage power supply from an alternating source, the amount of ripple can be further reduced by using larger value capacitors but there are limits both on cost and size. For a given capacitor value, a greater load current (smaller load resistor) will discharge the capacitor more quickly (RC Time Constant) and so increases the ripple obtained. Then for single phase, half-wave rectifier circuits it is not very practical to try and reduce the ripple voltage by capacitor smoothing alone, it is more practical to use "Full-wave Rectification" instead.

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. To overcome these disadvantages a number of Power Diodes are connected together to produce a Full Wave Rectifier.
The above information are extracted from http://www.electronics-tutorials.ws/diode/diode_5.html

The following applet simulate the effect due to different R-C values.
The blue curve is the input signal, the red curve is the output. Enjoy it :)