the voltage follower
The voltage follower is a non-inverting amplifier with a gain of exactly 1, so the output simply copies the input. At first this sounds pointless, output equals input, but it is doing something valuable: it transforms impedance. It presents a very high resistance to the source and a very low resistance to the load, acting as a sturdy buffer between a weak source and a heavy load.
It is built by wiring the output directly back to the minus input, which is the case Rf equals 0 and Rg is infinite, giving gain 1 plus 0 equals 1. By the virtual short, the minus input equals the plus input equals Vin, and the minus input is the output, so Vout equals Vin. The plus input draws almost no current (megohms and up), yet the output can drive a low-impedance load stiffly, sourcing milliamps with no sag.
This solves the loading problem that wrecks naive designs. A divider made of two 100 kilohm resistors gives a nice 1.5 V reference, but it sags the instant you draw current from it; put a follower after it and you get a stiff 1.5 V that can deliver real current. Buffers appear everywhere, isolating a high-impedance stage from whatever comes next so the two stages do not disturb each other.
A 100 kilohm and 100 kilohm divider across 3 V gives 1.5 V but collapses if a load draws current. Feed that 1.5 V into a voltage follower and the output stays 1.5 V while sourcing milliamps to the load.
No voltage gain, but it turns a fragile reference into one that can drive a load.
It gives no voltage gain. Its job is impedance transformation (current gain), not amplification, and it still cannot exceed the op-amp's output current or rail limits.