ok, k theory, let's see, as i recall grothendieck introduced a group built on vector bundles, on a manifold. one has an operation on bundles by tensor product. this let's us regard them as basically a monoid like the natural numbers. then we can imitate the procedure of constructing all integers from the natural numbers, to construct a group whose "positive" elements are the vector bundles. this is called the K group on that manifold.
Correction: the operation is direct sum, not tensor product.
I'm no expert on K-theory, either, but I know a thing or two about it. This might get technical in places, but maybe Mathwonk and morphism will be interested if nothing else, and it's good for me to try to summarize what I know to clarify the ideas for my own sake. You can always look at Wikipedia for definitions. Also, I may get sort of speculative as I go on because my knowledge will peter out pretty quickly.
If you want it to be a group, somehow you have to find a way to get an identity element and inverses. That doesn't look like it's going to happen if you just take vector bundles with direct sum because when you direct sum them, they just get bigger and bigger. So, what you have to do is consider stable equivalence classes of vector bundles--i.e. you declare the vector bundles to be equivalent if they become isomorphic after direct summing with a big enough trivial bundle. Any vector bundle over paracompact base is trivial after direct summing with some other vector bundle (by a trick kind of like Whitney embedding theorem or for bundles over manifolds, you could use the Whitney embedding theorem itself).
So, that's topological K-theory, and it was Atiyah and Hirzebruch who started the subject. Actually, Grothendieck started algebraic K-theory before the topogical K-theory, and Atiyah and Hirzebruch were inspired by that. The algebraic version is K^0 of a ring.
For a geometrically-minded guy like me, the next place to look would be simple homotopy theory, which is concerned with K^1 of group rings. Simple-homotopy theory is sort of a cell-by-cell geometric approach to homotopy theory. If you have a CW complex (actually, Whitehead invented CW complexes in this context because of their technical advantages over simplicial complexes), you might wonder if you can do the homotopies one cell at a time. It turns out you can't do it in general, but the obstruction to being able to do it is determined by groups called Whitehead groups. There's a geometric definition of the Whitehead group of a CW complex, but also an algebraic one as K^1 of the group ring of the fundamental group (actually, maybe it's the reduced K^1, which is a quotient of K_1 by something). The place to read about this is Cohen's book, Simple Homotopy Theory. Very nice and well-motivated book. Before Whitehead torsion, there were other torsions, like Reidemeister torsion, which was introduced in order to classify some 3-manifolds called Lens spaces. So, the theory of K^1 goes back to the 30s, I guess, although it wasn't called K_1 until while later.
So, apparently there is some kind of relation between K^0 and K^1 of a ring. Some kind of similarity. I'm not quite sure what it is. It may be cheating to say K^0 and K^1 at this point because I doubt they were called that originally. But, anyway, someone noticed some similarity there, and with that in mind, Milnor defined another group, K^2. And then the question arose as to whether K_0 and K_1 were part of a sequence, K^n. My guess is that this was conjectured with classifying spaces and cohomology theories in mind.
Given a ring, you can form the group of n by n matrices over the ring, GLn(R). And there's an inclusion map from GLn(R) into GLn+1(R). If you keep going and take the union of all those (direct limit), you get a big group called GL(R). Given a group, you can form a classifying space, BG, and the bundle EG over it. The significance of BG is that you can get any principal G-bundle by mapping the base into BG and pulling back EG. Homotopy classes of maps into BG correspond to isomorphism classes of principal G-bundles.
So, for example, homotopy classes of maps into BGL(R) give principal GL(R) bundles. So, it looks like that ought to be related to K groups of a space, somehow, since maybe GL(R) could act as symmetry groups for stable equivalence classes of vector bundles if R is a field, for example. So, building on those kinds of ideas, Quillen introduced his Quillen-plus construction. The plus construction was method of Kervaire for modifying the fundamental group without changing homology and cohomology, but Quillen applied it to BGL(R). I guess you just attach some cells to it. I'm not sure what the accomplishes, but evidently, it's pretty important, since he won a Fields medal for it. So, to define K^n, you take the nth homotopy group of the Quillen plus contruction.
If it were just BGL(R) without the plus construction, it's giving you principal GL(R) bundles over spheres, except that that would correspond to free homotopy classes of maps of spheres into BGL(R), whereas the homotopy groups are maps with basepoints. The point of the plus construction seems to be to kill off the commutator subgroup of the fundamental group, so you have free homotopy classes, eliminating the basepoint dependence.
Somehow, this is supposed to be related to some algebraic geometry and number theory stuff that I know almost nothing about.
Topologically, again, I'm guessing maybe if you want to deal with stable bundles and want to study them in a cell by cell way, perhaps, you would care about maps from spheres into the Quillen plus construction. If you want to look at one cell, any bundle is trivial over the cell, but you do care about the how the bundle over the cell is glued to the rest of the bundle. So, I could see it being relevant there. That may be what K theory as a cohomology theory is telling you about (there are K groups associated to rings and K groups associated to spaces which form a cohomology theory).
Stable bundles are the kind of thing maybe a high-dimensional topologist might care about, so I could see it coming up in surgery theory or something (took a class from a surgery theory guy who is interested in algebraic K-theory). And also, of course, algebraic topologists would be interested, too. High dimensional topology is very homotopy-theoretic now due to surgery theory tools that boil a lot of it down to homotopy theory.
It always seemed like kind of an obscure subject to me. I went to a talk about it last year and it was pretty much over my head. But after writing this and putting some strands together, it actually seems like there's something pretty cool going on there.
