A discrete-time LTI filter can be classified by the duration of its impulse response.

A finite impulse response filter, or FIR filter, has

$$h[n]=0$$

outside a finite set of indices. Its output can therefore be computed from a finite weighted sum of input samples:

$$y[n]=\sum_{k=0}^{M}b_kx[n-k].$$

An infinite impulse response filter, or IIR filter, has an impulse response that continues indefinitely. A common implementation uses feedback from previous outputs:

$$y[n]=\sum_{k=0}^{M}b_kx[n-k]-\sum_{k=1}^{N}a_ky[n-k].$$

FIR filters can be made exactly linear-phase and are automatically BIBO stable when their coefficients are finite. IIR filters can achieve sharp frequency selectivity with fewer coefficients, but their poles must be placed so the recursion is stable.

FIR and IIR describe system structure, not whether a filter is low-pass, high-pass or band-pass. Frequency selectivity and impulse-response duration are separate classifications.