Think of a filter as an electronic circuit that has a voltage input and a voltage output. In control theory a filter is described by how it reacts to an "impulse", which is essentially a brief peak followed by zero voltage.

Finite Impulse Response means that the output of the filter will eventually go to zero - after a finite duration. Infinite Impulse Response means that the output of the filter may never go to zero.

It can be proved that FIR and IIR filters have different properties; the article mentions that IIR filters are necessarily stateful, for instance.