Most wood stove fans rely on something called the Seebeck effect. The Seebeck effect is the inverse to the Peltier effect, and the two are often discussed in tandem.

In lay terms you can draw a small amount of voltage from the movement of heat. On a woodstove this is accomplished by having a large surface area in contact with the metal of the wood stove. The top of the stove fan is usually thinner, with fins or a grid to assist in drawing up and radiating the heat away from the surface of the stove. In the middle of this temperature gradient is a wafer of material with wires inside of it, known as a thermocouple. As the heat is conducted from the stove and up the stove fan the heat gradient produces a small amount of electricity that is captured in the thermocouple and transferred to a small motor that powers the fan blades.

The process is wildly inefficient for most other practical applications, however for something like a woodstove the intent is to direct some of the radiant heat forward via convection. Any loss in efficiency is in the form of radiant heat, so it's not a loss in the sense that the stove is still heating/warming a given area.