Why do we care about equivalence relations?

In the context of the connected components of a graph, the idea isn't very interesting, because the picture for connected components of a graph is already so clear and easy to understand geometrically.

But one of the principal techniques of mathematics is to find the underlying structure in one context, and then try to make an abstract model of it that might apply in other contexts as well. By doing this, we understand the connections between the two contexts, and we can sometimes solve difficult problems in one context by applying tools imported from some other context.

We don't formulate the idea of a partition or an equivalence relation because we want to study the connected components of graphs. For that it's unnecessary, because graph components are simple.

But that simplicity makes graph theory a good place to start understanding the idea of an equivalence relation, so that when you happen across it in a different context, where it might be more useful, you can recognize it and say “oh, we can model this with an equivalence relation, which means that it partitions the structure into components, and I already know some theorems about how that will work and some techniques I can use.”

And you have a language for talking about these things, which can be applied in many different situations, so that when you say “consider the equivalence classes of (something) under (some relation)” you and the people you are talking to instantly get an idea of what is going on: these classes are disjoint, every object is in exactly one class, and so on, just like the components of a graph.

Examples

1. What are fractions? They are notations of the form $\frac ab$, where $a$ and $b$ are integers, and $b$ is not zero. What are the rational numbers? Just fractions? No, they are the equivalence classes of fractions under the relation that says $\frac ab \equiv \frac cd$ if $ad=bc$, because for example $\frac 36$ and $\frac8{16}$ are the same rational number. Okay, big deal, we already know all about rational numbers. But having identified the process, we can now apply it to all sorts of things more complicated than the integers. Can we do the same construction for, say, polynomials? (We can!) Are the results useful and interesting? (They are!)

2. We can relate the complex numbers to polynomials by defining each complex number as a part of a partition induced on the set of polynomials by a certain equivalence relation. By using different equivalence relations we can define different systems analogous to the complex numbers and use them to study properties of polynomials.

3. Mathematics has a structure called a group, which is a model of a way in which a thing can have symmetry. There is an important “quotient” operation on a group which arises when you consider certain symmetries to be “equivalent”; the quotient is a group that describes the symmetries of the resulting equivalence classes.

4. This is a very specific example: Consider some geometric object which has an even number of symmetries. Then the object must have at least one symmetry which is an “involution”: this means that if you perform the symmetry exactly twice, the object is back in its original position. Objects without involutions must have an odd number of symmetries! (An example is a table-saw blade with 37 teeth.) I hope this is not obvious! But it is very easy to show if you consider the right equivalence relation on the symmetries.

5. An important technique in physics is to analyze the symmetries of the universe itself. For example, the laws of conservation of momentum and energy are consequences of certain symmetries of space-time. The abstract structure of spaces with these symmetries is often best understood as a particular quotient group.

6. This “quotient” idea applies in other situations too. Many kinds of mathematical structures are best understood as quotients, in this sense. For example, in topology we often view a circle as being a quotient of a line segment, under the equivalence relation that says that the two endpoints are equivalent. When we want to deal with a Möbius strip, we often formulate it as a certain quotient of a rectangle, where the equivalence relation makes certain points on the edge of the rectangle equivalent. Many related objects, much weirder than the Möbius strip, can be dealt with similarly.

We don't study the identification between partitions and equivalence relations just because it is cool. It is also useful in understanding other things. Partitions and equivalence relations pop up everywhere. Sometimes the partition is obvious, but the equivalence relation is easier to understand; then it is helpful to reinterpret the partition as an equivalence relation. Sometimes it is useful to go in the other direction instead.

One thing I haven't seen mentioned yet: Equivalence relations give us a tool for removing distinctions we are not interested in, so that the ones we are studying are all that remain.

For example, in topology it is very common to construct a new topological space by gluing existing ones together. MJD touches on this, but didn't show the power. Look at a square sheet. I would like to glue the two sides together:

How do you do that in topology? By equivalence relation! If $S = [0,1]\times[0,1]$ is the square, then I define $$(x_0,y_0) \sim (x_1, y_1) \iff y_0 = y_1 \text { and } (x_0 = x_1 \text{ or } |x_0 - x_1| = 1)$$ Points in the middle or top and bottom of the square are equivalent to only themselves, but points on the LH side is equivalent to points on the RH side at the same height. I drop to the set of equivalence classes $S/\sim$, and note that it has been proven that topologies are compatible with any equivalence relation, and I'm done! Just by putting together that equivalence relation, I've defined a cylinder complete with topology. But that isn't the end. With just a little twist, I get something different: $$(x_0,y_0) \sim (x_1, y_1) \iff (x_0,y_0) = (x_1,y_1)\text{ or } (|x_0 - x_1| = 1 \text{ and } y_0 = 1 - y_1)$$ Now the space of equivalence classes is the Möbius strip, completely defined as a topological space by that little bit.

In the original square I had distinctions that were not desirable (LH side vs RH side). So I created an equivalence relation that ignored those distinctions. By passing to the equivalence classes, I have now gotten rid of the unwanted distinctions while keeping what is useful.

I can also glue the top and bottom sides together:

• if both top and bottom are glued straight and right and left are glued straight, you get the torus.
• if the top and bottom are glued straight, but the right and left are glued twisted, you get the Klein bottle.
• If both top and bottom and right and left are glued with twists, you get the projective plane.

And if you just make all boundary points equivalent to each other, then the entire boundary collapses to a single point, and you get the sphere.

This is just a tiny taste of the power of this technique. Every compact surface without boundary can be built by cutting a bunch of holes in a sphere, and then to each hole, either glue a torus with a single hole in it (gluing the borders of the two holes together), or else glue the border of a Möbius strip to the border of the hole.

Similar examples can be found in any field of mathematics. I was originally going to use building the tangent space in differential geometry, but switched to surfaces as being more accessible.

Equivalence relations are one of the most powerful tools in the mathematician's tool box. You will find them used everywhere.

One instance in math where the notion of an equivalence relation is particularly useful, and makes the axioms of an equivalence relation seem natural, is considering a set modulo some equivalence relation.

It is also useful when you have some sets with some type of structure (groups, rings, topological spaces) and you want some way to think of them as being the same. We usually do this by defining an appropriate 'structure preserving function' between them and intuitively these functions should satisfy the axioms of an equivalence relation. Otherwise, what use would it be to have 3, say topological spaces, A,B,C such that we regard A and B as the same B and C as the same but not A and C as the same.

In short an equivalence relation is a nice way to make precise the manner in which we can regard two things as the same (even if they are not "equal"). For the laymen example suppose I am talking about all the shoes I own, but I don't want to talk about distinct shoes, I just want to talk about all the pairs of shoes I own. My blue nike that goes on my left foot is not actually the same shoe as my blue nike that goes on my right foot, but if I consider my shoes modulo the equivalence relation that they belong to the same pair - both of my blue nikes just became the same shoe. In essence I can talk about, or prove things about, both shoes by just considering one of them. Because modulo the equivalence relation, they are "equal".

Edit: added after original post. Also, something I noticed as I was advancing into higher more abstract mathematics (which I am still doing!) was a common theme of certain math concepts. We love to take some normal concept such as 'equality' or 'distance' and say 'what are the properties that are inherent to distance' or 'what are the properties that are inherent to the notion of equality'. If you sit down and brainstorm for each, you will likely come up with the axioms for a metric space in the distance case, and the axioms of an equivalence relation in the equality case. Then, mathematicians love to say 'what happens in the more general setting where more general objects obey these properties?'. In this sense, the "motivation" would be more along the lines of curiosity. Then, if the theory is turning out to be fruitful and producing interesting results, that further motivates the study of this new 'generalized' notion. I actually haven't studied graph theory, so I can speak to much about your example regarding connected components, but my hunch is that using the equivalence relation is 'just general enough' in the sense that, by avoiding using an equivalence relation you could, at some point, be limiting your self by not being general enough or get wacky results because things are too general. Me personally, everything I study is algebraic - and equivalence relations are imperative. They help to study smaller algebraic structures without forgetting the old structure it came from - and again help make precise what it means for two algebraic structures to be equal.