Full Idea
A relation R on a non-empty set S is an equivalence relation if it is reflexive (for each member a, aRa), symmetric (if aRb, then bRa), and transitive (aRb and bRc, so aRc). It tries to classify objects that are in some way 'alike'.
Gist of Idea
Equivalence relations are reflexive, symmetric and transitive, and classify similar objects
Source
Seymour Lipschutz (Set Theory and related topics (2nd ed) [1998], 3.9)
Book Reference
Lipschutz,Seymour: 'Set Theory and related topics (2nd ed)' [McGraw-Hill 1998], p.73
A Reaction
So this is an attempt to formalise the common sense notion of seeing that two things have something in common. Presumably a 'way' of being alike is going to be a property or a part