Convolution explains the connection between input signal ( $x[n]$ ) ,impulse response ( $h[n]$) and the output signal ( $y[n]$).

          $y[n] = x[n]*h[n]$

Convolution of 1-D signal can also be expressed as

         $ y[n] = \sum_{k=-\infty}^{+\infty} x[k] h[n-k]$

Discrete convolution can be expressed as

       $y [n] = \sum_{k=max(n-M,0)}^{min(n,K)} x[k] h[n-k]$

In the above expression $x[n]$ and $h[n]$ are signals having finite length of K+1 and M+1 respectively resulting in a finite signal of length K+M+1.