There are two types of digital filters, commonly called IIR and FIR, which stand for Infinite Impulse Response and Finite Impulse response respectively. What this definition means it that if you ping an IIR filter, as the response decays it will asympotitically approach zero but never quite reach it (e.g. 0.1, 0.01, 0.001, 0.0001, etc., etc. but never exactly 0). The FIR filter, on the other hand, will at some point reach zero. Practically, what this means in the IIR filter uses feedback- the output at time t is dependent on the input at time t as well as the output at time t-1 but the FIR filter is only dependent on the inputs.
We care about this because digital math can never have infinite precision - there is always some rounding error. Feedback can tend to compound rounding errors. Thus IIR filters can sometimes be numerically unstable. On the other hand, FIR filters are always stable. Another advantage of FIR filters is that they can have linear phase. The disadvantage is that an FIR filter generally requires a higher filter order to achieve the same performance (i.e. more multiplications and additions).
