What is filter memory? I have just started learning some DSP, I keep seeing things like memory in regard to filters.  I am unsure exactly what this means and why it is useful.  I understand that increasing the number of taps increases the memory etc.. and a tap is essentially just a coefficient for changing the signal.
Filter memory refers to the effect of past input on the current input. Infinite impulse response (IIR) filters and finite impulse response (FIR) filters are fundamentally different from the perspective of their memory. The memory of an FIR will essentially be the number of taps, while IIR has theoretically infinite memory (although when they are stable, the memory passes under the numerical accuracy of the system beyond a certain point).

More on this http://gregstanleyandassociates.com/whitepapers/FaultDiagnosis/Filtering/Filter-Memory/filter-memory.htm.