Discrete Convolution and Сross-Correlation. Theory and Implementation in Python

What are convolution and cross-correlation? How can we implement and apply them? Often we need to express the interaction between two processes represented by some functions. For example, let us find out how the amount of snow is changed over time in a given area. But snow does melt and it depends not only on the current total amount, but also it depends on the time when this snow fell.