Convolutional Neural Networks (CNN) are mainly used for image recognition. The fact that the input is assumed to be an image enables an architecture to be created such that certain properties can be encoded into the architecture and reduces the number of parameters required.
The convolution operator is basically a filter that enables complex operations to be performed on an image. Examples are edge detection, gradient recognition and smoothing. This allows pertinent data to be extracted from the image.
Mathematically, a convolution is an operation on two functions producing a third function which is an integral that provides the amount of overlap of one function as it is shifted over another function.