plus construction
Sometimes a space has too big a fundamental group to be useful, yet you do not want to disturb its homology. Quillen's plus construction is a surgical tool that kills a chosen well-behaved chunk of the fundamental group while leaving all homology exactly as it was. It is the bridge that turns the group-theoretic data of GL(R) — a group, hence a space with no interesting higher homotopy on its own — into a space whose higher homotopy groups are the higher K-groups.
Let X be a connected CW-complex and let P be a perfect normal subgroup of pi_1(X). The plus construction produces a space X^+ together with a map X -> X^+ such that pi_1(X^+) = pi_1(X) / P and the map induces an isomorphism on homology with any local coefficient system pulled back from X^+. Concretely one attaches 2-cells to kill P (changing homology) and then 3-cells to repair the homology damage; perfection of P is exactly what makes this repair possible. The result X^+ is unique up to homotopy equivalence under X.
Applied to X = BGL(R), the classifying space of the infinite general linear group, with P = E(R) the elementary subgroup (which is perfect by Whitehead's lemma), one gets BGL(R)^+. Quillen defines K_n(R) = pi_n(BGL(R)^+) for n at least 1. This recovers K_1(R) = pi_1 = GL/E and K_2(R) = pi_2 = the Schur multiplier of E(R), and produces all higher K-groups uniformly. The same idea, applied to the group completion of a symmetric monoidal category, gives K-theory of categories.
For R = Z, pi_1(BGL(Z)^+) = K_1(Z) = Z/2Z and pi_2 = K_2(Z) = Z/2Z, while pi_3 = K_3(Z) = Z/48Z. The plus construction has turned the homology of GL(Z) — a hard but in-principle accessible object — into these homotopy groups, the higher K-theory of the integers.
K_n(Z) = pi_n(BGL(Z)^+): K_1, K_2 = Z/2Z, K_3 = Z/48Z.
The space BGL(R)^+ is an infinite loop space, so the K-groups assemble into the homotopy groups of a spectrum, the K-theory spectrum K(R). This extra structure is invisible from the plain plus construction but is what makes K-theory a generalized cohomology theory with products and operations.