A filter can be literally any operation on a signal! But the most common use of the term describes devices that boost or attenuate regions of a sound spectrum, which is the usage we are generally concerned with in audio engineering usage.

Digital filters work by using one or both of two basic methods:

a) Delaying a copy of an input signal slightly (by one or several periods) and combining the delayed input signal with the new input signal

b) Delaying a copy of the output signal and combining it with the input signal