Whenever you connect something (a "load") across a voltage source, obviously some current will be pushed through that load.
By "voltage source", I mean anything that produces a potential difference, whether it be a battery, a solar panel, a microphone, an audio amplifier output, or the output from an op-amp. Literally anything that produces a potential difference. Usually, in the context of a buffer, the source is some kind of time-varying potential difference, produced by an amplifier stage of some kind, such as an op-amp, or by a transducer such as a microphone.
All voltage sources have some inherent output resistance, via which this current must pass in order to reach the next stage. Output current will cause a voltage "drop" across this resistance, according to Ohm's law \$V=I\times R\$. Consequently, the next stage (the "load") never sees the source's actual potential difference, only what's left over after the source's own internal resistance has "consumed" some fraction of it.
Obviously this is inconvenient, because it's a cause of signal distortion. To mitigate this distortion, you have three options:
Design a better source, with lower output resistance
Design a better load, that draws as close to zero current as possible
Put something between the two stages, a "buffer", that draws almost zero current from the source, and reproduces source's potential at its own output, having a low enough resistance that the load can draw whatever it needs without significantly affecting signal potential.
Option 3, the buffer, has two important properties:
Very high input resistance, so that it draws as little current from the source as possible.
Very low output resistance, so that the next stage can draw whatever current it requires without compromising signal potential.
In terms of your question, which refers to "current gain", these two properties could be paraphrased as a buffer having a large current gain. That is, drawing insignificant current itself from the source, and providing any amount of current required by the next stage.
Since the intended purpose of a buffer stage is usually to exactly reproduce some potential, in that context voltage gain is typically 1. However, there's technically no reason why you couldn't build a buffer with any voltage gain you desire. It might not be a "buffer" in the truest sense, but it can still perform the intended function of a buffer, which is to prevent subsequent stages from over-loading some weak voltage source.