the emitter follower

The emitter follower is the BJT's buffer — a circuit that gives almost no voltage gain but enormous current gain. Why would you want gain of 1? Because its job is not to make a signal bigger but to make it STRONG: it takes a delicate, high-impedance signal that cannot drive a heavy load and re-presents the same voltage from a low-impedance, muscular output. Think of it as an impedance transformer: weak in, same voltage out, but now able to push real current. It is also called the common-collector configuration, because the collector is the common terminal tied to the supply.

The signal goes in at the base and out at the emitter, and the emitter voltage simply FOLLOWS the base voltage, sitting one base-emitter drop (about 0.7 V) below it — that constant offset is why the voltage gain is just under 1, not exactly 1. The real value is in the resistances. The input resistance is high: looking into the base you see roughly beta times the emitter resistance, so a 1 kohm load looks like maybe 100 kohm to the source. The output resistance is low: it is approximately the source resistance divided by beta, plus the internal re, often just a few ohms. Worked feel: drive it from a 10 kohm source and the output resistance might be only 100 ohm — a hundredfold improvement in drive strength for free.

This is exactly what you put between a high-impedance stage (like a sensor or a common-emitter collector) and a low-impedance load (like a speaker, a long cable, or the next stage's input) so the first stage is not dragged down. It is the output stage of nearly every op-amp and power amplifier, usually doubled up as a push-pull pair. The honest limits: it gives no voltage gain, the 0.7 V offset shifts the DC level, and like any follower it can ring or even oscillate when driving a capacitive load if you are careless — a small series resistor at the output usually cures that.