Half-power point
The half-power point or half-power bandwidth is the point at which the output power has dropped to half of its peak value; that is, at a level of approximately -3 dB. In filters, optical filters, electronic amplifiers, the half-power point is a commonly used definition for the cutoff frequency. In the characterization of  antennas it relates to position describing directionality.

Amplifiers and filters
This occurs when the output voltage has dropped to 1/ (~0.707) of the maximum output voltage and the power has dropped by half. A bandpass amplifier will have two half-power points, whilst a low pass amplifier or a high pass amplifier will have only one. 

The bandwidth of an amplifier is usually defined as the difference between the lower and upper half-power points. This is therefore also known as the 3 dB bandwidth.  In the case of a low pass amplifier, there is no lower half-power point so the bandwidth is measured relative to direct current, i.e. 0 rad/s.

Antennas
In antennas, the expression half-power point does not relate to frequency: instead, it describes the extent in space of an antenna beam.  The half-power point is the angle off boresight at which the antenna gain first falls to half power (approx. -3 dB) from the peak. The angle between the  points is known as the beamwidth.