Grasping the concept of equivalence classes more concretely I have read about equivalence classes a lot but unable to understand exactly what they are. I know about equivalence relations, but I am having a hard time understanding what equivalence classes are exactly.
Can anyone explain the concept of equivalence classes clearly? 

The main thing to understand is that when you have an equivalence relation  $R$ given on a set $S$, you're able to create a partition on the set $S$.
Having a detailed look at the demonstration of this fact is a great move to understand the link between an equivalence relation and its classes.
