What is the exact meaning of unstable system in DSP? In physical systems I understand what is the meaning of stability or unstability. An operational amplifier for example, if working in positive feedback will either saturate or start osscilating (i.e will not have any stable state). thats clear to me.

But I am unable to understand what exactly we mean when we say an IIR filter (or any other digital system), for example, can become very unstable. 


What exactly inside the Digital Signal Proccessor  happens, what happens to the output physically? 
What exactly we mean by unstable system in this context?
Unstable typically means and unbounded output for a bounded input. In other words the output of, say , a filter can get infinitely large although the input is perfectly okay and of "normal" size.
A simple example would be the difference equation $y[n] =  x[n] + y[n-1]$. If we calculate the step response, i.e. $x[n] = u[n]$, we get y[0] = 1, y[1] = 2, y[2] = 3 ...
The output grows infinitely even though the input is a perfectly well behaved signal, bounded by 1.