From an implementation point of view there are two kinds of digital filter : 

The IIR filter calculates its outputs based on both its input and past values of its output, while the FIR filter only operates on its input signal. 
So what's an impulse? 
An impulse is a signal which has a value of zero everywhere except for a single sample which has a value of one. 
The impulse response of a filter is the output of a filter when the input to the filter is an impulse. 
FIR filters have a finite impulse response because any output sample is only calculated from a finite section of the input signal. 
IIR filters on the other hand feed their output back to their input and hence (theoretically) have infinitely long impulse responses. 