Antenna Gain explanation
Antenna efficiency is generally taken as the amount of total power radiated (in all directions) divided by the electrical power put into the antenna. It is not related to antenna gain.

Antenna gain is something a little more complex. A mathematical device called an "isotropic antenna" is a theoretical antenna that emits power equally in all directions. This is the base line for a lot of comparisons between antennas. No such antenna exists but it is still a good device for making comparisons.

An antenna that bunches the emitted power up so that it produces more watts/sq meter in one direction compared to a different direction is generally specified as having gain. A dipole antenna will concentrate power like this: -



It has a gain that slightly exceeds (by 2.15dB) the 0 dBi mark on the polar diagrams. Note I used the term "dBi" to indicate gain relative to the theoretical isotropic device.

So, an antenna with "gain" must has directionality - it forces more watts per sq meter in some directions and fewer watts per sq metre in other directions. 

Antenna gain works equally in transmit as it does receive - an antenna with higher gain will collect more watts and convert them to more volts in its "optimum" direction than it does in other directions. 

A receiving antenna has, what is known as, an aperture (think of it as a net which it catches radiated power - the bigger the net the more power it catches).

Both efficiency and gain need to be considered together though - an antenna with significant gain may still not be a good emitter if the efficiency is poor.