What is "gain" a measurement of in electronics?
Gain is just the ratio of two quantities. When talking about circuits, it's almost always the ratio of some measurement of the output of a circuit or component to its input.

As Olin said, the most common case is the voltage gain. That is, the ratio between the voltage amplitude of the output signal to the input signal.

Another common specification is power gain, the ratio of the output power to the input power. 

But some circuits are best characterized by a current gain. Some (transimpedance amplifiers) are best characterized by a transimpedance gain --- the ratio between the output voltage and the input current.

Another common usage of the term gain is in antennas, where the gain of an antenna is "the ratio of the power produced by the antenna from a far-field source on the antenna's beam axis to the power produced by a hypothetical lossless isotropic antenna, which is equally sensitive to signals from all directions." (per Wikipedia)