[datenwolf]

The term "gain" is a bit unlucky, because it does not really describe what's happening. A much better term would be "directivity". Antenna gain is always with respect to a reference pattern; usually a perfectly omidirectional isotropic pattern is assumed and the gain tells you, how much of the antenna radiation pattern is "compressed" into the spatial direction you're looking at, compared to a plain isotropic antenna.

For example if you're looking at a plain old dipole antenna, then in the plane perpendicular and centered around the dipole you'll see a +5dB gain as if the antenna was perfectly omnidirectional. But if you move out of the plane, say right into the axis of the antenna (but some distance away) there will be very few signal, so what would have gone there has been redistributed into a different spatial direction.

Now consider you have a wireless device sending with 20dB(mW) = 100mW and attach an antenna with a directivity gain of 20dB to it. Then to a receiver that happens to be located within the directivity pattern it looks as if there was a transmitter sending with (20 + 20)dB(mW)=100*100mW = 10W through an omnidirectional radiator.

It's one of the few cases where English is lacking a distinctive term. In German the term used for antenna gain is called "Gewinn", whereas amplifier gain is called "Verstärkung".