This is a very large subject; millions of people are engaged in the pursuit of working with clay, glaze, underglazes to create works of art and or craft (words that are endlessly debated, in the attempt to describe what is made).
There are three main areas defined by temperature range: lowfire("earthenware"), mid-range("low-stoneware"), and high-fire (stoneware and porcelain). Each has its own clays and glazes which are commonly used. Also, divisions are made according to whether works are "functional", meaning to be used in the home for dishes or vases, etc., or sculptural.
Pottery or ceramics is commonly taught in secondary schools and colleges and universities, and in art and craft centers in communities. It is one of the world's oldest art forms.