What is the difference between continuous and discrete convolution?

A convolution is a function defined on two functions f(.) and g(.). If the domains of these functions are continuous so that the convolution can be defined using an integral then the convolution is said to be continuous. If, on the other hand, the domaisn of the functions are discrete then the convolution would be defined as a sum and would be said to be discrete. For more information please see the wikipedia article about convolutions.