A voltage follower circuit produces an output voltage which is the same as the input voltage, so the output 'follows' the input – hence the common circuit name.
To understand the circuit operation, firstly notice the feedback connection linking the output (Vout) to the inverting input terminal (V−), hence causing these two voltages to be equal. A common op-amp assumption is to take the open loop gain (AOL) of the op-amp to be infinite, so V+ = V−, which in turn means that Vout = Vin.
This circuit uses negative feedback in which the fed-back signal tends to make the input smaller. Don't make the mistake of connecting the input voltage to the inverting input as this will result in positive feedback, causing the circuit to be unstable.
You might wonder what is the point of this circuit, when the output could simply be connected to the input by a piece of wire? To answer this question, it is important to understand that the op-amp offers a very high input impedance (Zin) and a correspondingly low output impedance (Zout), and the use of negative feedback further amplifies these values. This causes the circuit to act as a buffer, leading to its alternative name of a unity gain buffer.
The output voltage will accurately follow the input as long as the input stays within the limits imposed by the maximum output voltage swing, which in turn is related to the positive and negative supply voltages.