[QUOTE=Nava]
First it’s a branch of Logic then it isn’t… guess I’ll stick with using math to figure out how much solvent to put in paint and leave the definitions to those of you who actually get them 
(joking, joking)
[/QUOTE]
Ah, I see now, you may have been confused into thinking the kind of work done in set theory is the kind of work done in group theory and ring theory, possibly by these lines in the Wikipedia article on set theory: “Set theory is seen as the foundation from which virtually all of mathematics can be derived. For example, structures in abstract algebra, such as groups, fields and rings, are sets closed under one or more operations.” Those lines don’t mean that set theory is actually about the structures of abstract algebra. Rather, set theory is a subject of its own, about mathematical objects called sets, and not much else. (What are sets and what can we study about them? Well, I’ll touch on that in a second). But, in the current mathematical climate, set theory plays something of a foundational role for the other branches of math. A common viewpoint is that all the objects/structures that “exist” in the mathematical universe can be considered as particular sets; if you want to assume that some kind of mathematical structure exists, you consider this equivalent to the assumption that some particular kind of corresponding set exists, and then you depend on the set theorists to tell you that, yes, it’s ok to use such a structure, because they’ve established that such a set exists (generally from a particular theory of assumptions about the set theoretical universe, called ZFC).
That is to say, people from other branches of math will use the concepts of set theory to make the definitions at the bottom of their field, and depend on various assertions from set theory to justify their assertions that certain types of objects or structures exist. For example, if I’m doing work with the real numbers, I can start by writing up various axioms about the real numbers that I’ll use in my work (there are infinitely many of them, they have a linear order, every bounded increasing sequence of reals has a least upper bound, etc.); however, the nagging question will remain “How do I know that there actually is some structure of numbers which satisfies all these axioms?”. Or I might want to know whether a particular kind of ring exists, and be unable to come up with any examples or proofs of impossibility on my own. To resolve these existence questions, then, I can turn my descriptions of structures into descriptions of sets, and then ask “Does there exist a set like this in the set theoretical universe?”. And that’s the sort of thing that the axioms of set theory are designed to help me decide (“Yes, ZFC says it does”, “No, ZFC says it doesn’t”, “Hm, well, ZFC doesn’t seem to be enough for us to tell you, but if we add these other commonly studied set theoretic assumptions, we can get an answer”). However, once I’ve finished seeing that, yes, the work I’m doing makes sense, because the structures I’m working with can be interpreted as things in set theory, I generally stop caring about the minutiae of the set theoretical universe, and get back to business dealing directly only with the various properties I asked my structure to have (once I see that there is some set which can be interpreted in some way as the real line, I’ll stop worrying about the details of this interpretation, and just think in terms of real numbers again, secure in the knowledge that they can be made sense of in terms of sets if needed. I’ll know that the logical consistency of my work, if I’m asked to justify it, follows from the logical consistency of the set theoretic assumptions I grounded it in. And the gold standard of mathematics is to ground things in ZFC, so if I can pull that off, there’s no question anymore as to the legitimacy of my work.).
In practice, “core” mathematicians (i.e., those who don’t work in logic) just have some intuition about what is and what isn’t allowed, and implicitly assume that the work they’re doing can be formalized in terms of set theory and legitimized by the rules of ZFC, without ever doing so explicitly. Every now and then, though, they may be unable to decide if some particular sort of object exists (e.g., is there a function from the reals to the reals with such and such a property?) and have to go running to the set theorists.
Now, for things like calculus, you can mostly set it up once and be happy for life: show that the structure of real numbers exists, that we can make sense of notions like an infinite sequence of reals, a function from reals to reals, etc., and then once you’ve implemented all this set theoretically, you very rarely have to ask about existence in the set theoretical universe again; you can mostly get by with just the informal understanding of what kind of ways to construct functions, etc., are legitimate. However, for abstract algebra, which is more directly concerned with looking at a wide variety of structures and trying to find ones with special properties, the connections to existence questions from set theory are perhaps more explicit. And I think that’s what the Wikipedia article was trying to say. But, still, for the day to day stuff, even the algebraists can get by on just their intuition about what kind of constructions are legitimate and which aren’t.
Ok, that was long and rambling, and fairly dense, and probably shot to hell any reputation I have as a man of clarity and useful explanation.
Oh, what are these “sets” that everybody keeps going on about? Well, basically, a set is a collection of objects. So, something like {3, my house, George Bush} is a set, a collection of three objects. Or the set of all natural numbers. But, like I said, there’s the catch that we take the point of view that nothing exists except the sets. So {3, my house, George Bush} isn’t the sort of thing the set theorists will study; they’ll look at things like … the empty set with nothing in it, the one-element set containing only the empty set, the two-element set containing both of those. And then more exotic, but really useful things, like the infinite set containing all the sets that can be built up finitely from the empty set, or the even bigger infinite set containing all the sets whose members are drawn from that first infinite set. And so forth. There’s a bunch of rules in ZFC asserting that new and more exotic sets can be built up in various ways starting from the empty set (and also asserting that certain kinds of sets don’t exist at all), and what set theorists do is study the consequences of these rules for the nature of the set theoretical universe, and the consequences of various additions to these rules.
I could make a longer, more explanatory post about set theory and ZFC later, but that should be enough for now.
I probably will, though, after that rambling and dense explanation of the foundational role of set theory, you may no longer want me to. 
Sure, there are three big things that get called algebra. There’s normal high-school type algebra (elementary algebra), but nobody really does research work on that, that’s all known and settled. Then there’s abstract algebra, which is some sort of generalization of parts of elementary algebra; instead of being told the various identities that are legitimate for symbolic manipulation, and then using them to reduce or solve equations, you rather choose a bunch of function and constant names and what identities among them you want to consider legitimate, and then look at what properties follows from those identities, and what kind of structures can be made to have those identities. And then there’s the third big thing, linear algebra, which is concerned with vector spaces and linear transforms (which causes them to be concerned with work on matrices). This comes up a lot in physics, naturally, and permeates both work on relativity and work on quantum mechanics.
