In plain English
Classification is a machine learning task where a model assigns each input to one of a set of predefined categories. The model learns from labelled examples which features tend to go with which label, then applies that to new, unseen inputs. It is among the most common and useful things machine learning does.
A worked example
A spam filter classifies each incoming email as 'spam' or 'not spam' based on patterns it learned from millions of previously labelled messages.
Common confusion
Classification picks from fixed categories, which is different from generation. A classifier labels things; a generative model creates new content.