For me, one interesting thing about algebraic K-theory is L-theory (which I wish I understood better). This is in no way going to be coherent, but:
Let's say you are interested in classifying manifolds. That's not going to be possible, because any finitely presented group is realizable as the fundamental group of some 4-manifold, and "most" finitely presented groups don't have solvable word problem. OK then, the next best thing is to try to classify manifolds within a fixed homotopy type. Surgery theory is a technique for doing this. Given a homotopy type, you construct a CW-complex X with that homotopy type, and your first question is whether X is homotopy equivalent to a manifold. Roughly speaking, this is determined by the normal bundle. So for X to be homotopy equivalent to a manifold, you want there to exist a suitably defined bundle map $(f,b)\colon M \to X$,which is normal bordant to a homotopy equivalence. This latter condition (for a bundle map to be normal bordant to a homotopy equivalence) is detected K-theoretically (I don't understand this well enough to attempt to try to explain it).<br>
Note that homological techniques are sufficient for simply connected manifolds (Browder, Kervaire, Milnor, Novikov...), but are not useful for manifolds which are not simply connected. And this is where K-theory enters.<br>
The main point seems to be that algebraic K-theory provides the machinery to work with quadratic forms over nonabelian group rings with involution ($\mathbb{Z}[\pi_1(M)]$ in our case).<br>
See also <a href="http://www.math.uni-bielefeld.de/~fw/an_outline_how_manifolds_relate_to_algebraic_geometry.pdf">Walhausen's survery</a>, which I don't yet understand.