While Brian's link is as good of an answer as I would ever be able to give to the technical aspects, I would like to point out that equivalence relations and equivalence classes are one of the most common tools used in mathematics. We use them all the time to remove aspects of a construction we are not interested in, leaving only what we are after left behind.
A practical example from differential topology: Start with a square-shaped region of the plane, with vertices labeled A, B, C, D. This is an example of a "manifold with boundary", the boundary being the edges of the square. Our purpose to get rid of the boundary, leaving us with a compact (finite in size) manifold without boundary. We do this by building equivalence relations that preserve the structure we want (nearness) while getting rid of the boundary we don't.
I will produce 4 equivalence relations. Each gives us a different manifold. All four have this in common: every interior point of the square is equivalent only to itself. And every edge point is equivalent to exactly one other edge point, (except the vertices). How they differ is in how these points are stitched together. Denote the 4 sides by their endpoints: $AB, BC, CD, DA,$ or $BA, CB, DC, AD$. When I say that side $XY \sim WZ$, I mean that $X \sim W, Y \sim Z$ and any point on $XY$ a distance $t$ from $X$ will be equivalent to (and only to) the point on $WZ$ at a distance of $t$ from $W$.
- Relation (1) makes $AB \sim AD$ and $BC \sim DC$.
- Relation (2) makes $AB \sim DC$ and $AD \sim BC$.
- Relation (3) makes $AB \sim DC$ and $AD \sim CB$.
- Relation (4) makes $AB \sim CD$ and $AD \sim CB$.
For each relation, we form a new space out of the equivalence classes. The equivalence classes themselves are the points in our new manifold. Since in all relations, the interior points of the square are equivalent only to themselves, they each show up as a singleton set in our new space. Effectively, the interior points are unchanged. On the edges, the equivalent points are combined by the equivalence relation to form a single new point. These equivalence relations preserve the concept of "nearness" (which for simplicity's sake, I will leave as intuitive), so we can think of it as "bending the square over and gluing the edges together". Except that by using equivalence relations, we don't actually have to warp the square in space. We get what we need just from the logic of the relation.
Relation (1) gives us a surface that behaves exactly like a sphere. And for that reason, we call it the sphere.
Relation (2) gives us a surface that behaves exactly like a torus (surface of a donut shape).
Relation (3) gives us a surface called the Klein Bottle. It is unorientable (it has only one side, like a möbius strip), and does not exist in 3D Euclidean space, except as a self-intersecting surface.
Relation (4) gives us a surface called the Projective Plane. It is also unorientable and also does not exist in 3D Euclidean space, except by intersecting.
So by using these equivalence relations, I am able to construct these 4 surfaces in a form that allows me to see how they are related and how they differ. And to explore their properties by examining behavior on a square. Every subject in mathematics is replete with similar examples.