If an equivalence relation is defined on a set, then we can classify each element of that set using the relation, by putting all elements that are equivalent (according to the relation) in the same class and elements that are not equivalent (according to the relation) in different classes.
The properties of equivalence relations ensure that a given element will be in exactly one class. Therefore, an equivalence relation can be used to partition a set into disjoint subsets. These subsets are called equivalence classes.
For example, say our set is all the people in the world and our equivalence relation is "share the same birthday". Then this partitions the set into 366 equivalence classes. I would be in the equivalence class with all the other people born on 19th November.
UPDATE: next post
The original post was in the category: poincare_project but I'm still in the process of migrating categories over.