What exactly are equivalence classes What exactly are equivalence classes?  Suppose I have an equivalence relation $\sim$ on some set $X$ we denote this as $x \sim y$. The equivalence classes are then $[x] = \{y \in X : y \sim x\}$.  
However I have struggled as to what this actually mean, does this mean that for some $y \in X$ the equivalence classes are just the objects that satisfy the symmetric property?

Equivalence classes are sets of elements which are all equivalent between them. For instance if the equivalence relation $\sim$ is "having the same sex", then there are two equivalence classes in the world: boys and girls (if we forget the ambiguous cases).
In the same way, if the equivalence relation is "being born the same year", then each year yields a different equivalence class of all the people from this year.
To sum up, an equivalence relation cuts the universe into "potatoes" of elements: inside a potato, all elements are equivalent to each other, and a potato is called an equivalence class.
