What is the purpose of sets? Why do we use them? All is in the title. Why sets? Why do we need them and where are they important?
 A: Sets are useful because they are so incredibly general by their very nature. This is not surprising: what concept is more general and basic than the idea of grouping multiple things into an aggregate?
The starting point to any mathematics is the definitions, and without sets we would be hard pressed to give definitions of most mathematical objects (or rather, we would probably end up using sets implicitly in our definitions). Since we use them everywhere, we can unify virtually all of mathematics if we fix a constant notion of what a set is, thus providing a single theory underlying all of mathematics. If we did not formalize the notion of set, we would have a collection of mathematical theories all based on definitions implicitly involving sets, and students would be on math.SE asking if there might be some way to formalize the notion of set so that these intuitively obviously related definitions can be made rigorously related.
Here are some definitions relying on sets:


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*Functions. A function is basically a rule that maps numbers to other numbers. But what does "a rule" mean? It's quite vague - is $2x$ the same rule as $2x+1-1$? They're different rules in that they're calculated in different ways, but they always yield the same output. Hence we define a function as a set of ordered pairs (if you're studying set theory, I assume you're familiar with this definition).

*The domains of functions. So a function maps numbers to numbers, but the same concept of a rule that maps things to things pops up all over. Operators like the integral or the limit map functions to numbers, operations in groups map abstract group elements to other elements, other strange functions might map equivalence classes of infinite sequences to infinite sets of rational numbers, and so on. Again, these are all clearly basically the same idea: a rule that maps things to things. But you can't precisely pin down what "things" means unless you have a mathematical notion of an arbitrary set.

*How could we have things like algebraic structures or topological spaces without sets? An algebraic structure is a set together with operations on it. The whole power of modern algebra is that it can apply to a set of anything, as long as the operations obey certain rules. Ditto for topological spaces.


The last one is probably the most important. The power of modern mathematics is that we can build up theories that can apply to just about collection of things that obeys certain rules, but that's only possible if you have a mathematical notion of an arbitrary collection of things.
A: The purpose of sets is to house a collection of related objects.  They are important everywhere in mathematics because every field of mathematics uses or refers to sets in some way.  They are important for building more complex mathematical structure.  For example the ordered set $(x_1, x_2, x_3)$ of three points forms a vector in $\Bbb{R}^3$ if $x_i \in \Bbb{R}$.  I couldn't get through that definition without referring to another set $\Bbb{R}$. 
Sets can come with algebraic or topological properties that are useful.
A: The importance of sets is one. They allow us to treat a collection of mathematical objects as a mathematical object on its own right.
For dealing with finite collections of objects we can somehow wiggle around sets. We just specify our objects. So it's not hard to see that, for example, there is no one-to-one correspondence between a collection of five objects, and a collection of four objects. In fact you don't even have to use mathematics. Just cut off your right thumb1 and use your hands to see that.
However proving that you can write a bijection between the rational numbers and the integers is very unintuitive without the basic development of sets and set theory. Using this, for example, we can develop further objects, like constructing a function which is continuous almost everywhere, but its set of discontinuity points is a dense set.
Why is this important to begin with? Well, in an era long gone from our minds mathematics was without rigor. The concept of "function" was ill-defined. When it was somewhat better defined, people thought that all functions are continuous, and continuously differentiable, and so on. Or at least piecewise. Then counterexamples in the form of the Weierstrass function, and so on, which showed people how important is to have concrete definitions. 
Sets allow us to develop mathematical theories formally, by having the collections we want to talk about as mathematical objects on their own accord. Of course you can argue "Well, what if it wasn't called a set?", well in mathematics what matters is the definition, rather than the name. If you would have called it "Awesome Bag of Mystery Swag" and its properties were still the same as those of sets, then you would have had sets.
In fact a set is an awesome bag of mystery swag. It's a mathematical collection which "exists" - in the mathematical sense of the word. And once we realized that sets need some basic definitions themselves we wrote axioms and agreed that those are reasonable properties for sets to have.
Many people, of course, don't care and don't need axiomatic set theory2. Naive set theory, under the assumption that "almost every collection is a set, and surely every collection I care about is a set", works just fine for the general mathematician. And that's fine. But the ability, within a proof, to work with a collection of mathematical objects as a mathematical object itself is a wonderful thing. And that's the true importance of sets.
(And I haven't even began talking about how sets allow us to make everything into first-order using $\sf ZFC$ or some extension thereof, and how that really helps us out when we want to prove things; because that would require us to involve logic and model theory into the game. But that's a whole other point, which one might argue can be done without sets, somehow. Although I'm not sure how.)

Footnotes:


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*If you are already missing one finger, you don't have to do that; if you are missing more, cut the thumb of someone else's hand.3

*See the very recent discussion about naive and axiomatic set theories: Different types of Set Theory.

*Don't really cut anyone's thumb off!
A: Judging by your name, you are not afraid of a little bit (read:  a lot) of abstraction.  So, I'm going to address the spirit of the question while disagreeing with the underlying premise.
Sets were a useful artifice that may have outlived their role as the fundamental objects in mathematics.  They are a very useful category (and many other useful categories admit forgetful functors to Set), but it seems that the fundamental concept du jour is the topos.
John Baez has a gentle introduction to topos theory here.  A topos is type of category that has just enough of the structure of Set to be really powerful.
A: (Preliminary remark. There is an established subfield of mathematics called set theory. In the following I assume that your question is not referring to this field, nor to the logical implications of various axiom systems considered in set theory.)
The aim of any particular science is to bring "order", or "structure", into the things or phenomena observed. This is accomplished by inventing "notions" and observing or inventing "laws". 
Take biology as an example: In view of the living world around us one is quickly lead to distinguish "plants" and "animals". Later on one would recognize that the notion of "cell" is all important in understanding what is going on in the realm of life. "Understanding" means that the description of the totality we observe can be greatly reduced (in terms of kilobytes) by appealing to agreed on "notions" and "general principles" thought up underway.
The realm of mathematics is of course a spiritual one; but the same drive applies. Only late in the history of mathematics it became apparent that the notion of "set" is an extremely useful "Beschreibungselement" in dealing with mathematical phenomena.
A: The most important property of a set is that something is either in it or it is not in it. Symbolically, $x \in S$ or $x \not \in S$. 
Because of this, sets allow you to put all of the things you want to talk about into a sack that you can poke and prod at.  Using logic, intuition, and serendipity, you can discover interesting things about the things in that sack.
The importance of set theory can be misdirected. Instead of saying "the solution to the equation is $x = 7$ or $x=8$" you can now say "the solution set is $x \in \{7, 8\}$". That is just syntactical candy. Sweet but not mentally nourishing.  
However, when you get into things like algebra, topology, number theory, and so on. The language of sets makes it much easier to communicate what you are trying to say clearly and efficiently. Other theories, like topos and category theory are appearing and giving set theory some competition for the title of most useful and most funcamental. But set theory has been around much much longer.
