Finite Impulse Response vs. Infinite Impulse Response. IIR filters utilize feedback while FIR filters don't. If you put a string of 1s followed by 0s into an IIR filter, the output will keep on ringing (hence infinite). A FIR filter on the other hand eventually decays to 0 (hence finite). In practice, both decay to zero but FIRs decay faster than IIRs.
IIR filters are typically preferred in cost-sensitive applications because they usually require less memory to implement. But FIRs offer calculation advantages. FIRs can also be designed with steeper slopes than IIRs. Next generation electricity revenue meters for example will incorporate simple DSPs for computing various power and energy parameters but the meter business is extremely cost-sensitive. Engineers will most likely choose IIR filters to lower memory usage but they'll be pulling their hair out to do what they could with FIRs very easily.
Now you're going to ask me which is better for audio and I'd have to tell you that FIR theoretically offers higher performance but whether the FIR/IIR difference is audible or not is beyond my area of expertise. Also, it depends on the design. A junior can botch a FIR filter and a senior can work magic with an IIR filter. If anyone knows, chime in.
