K_1
If K_0 counts modules, K_1 measures the symmetries of free modules — the invertible matrices — but only after discarding the ones that any high-school student could obtain by elementary row operations. Over a field, every invertible matrix is a product of elementary operations together with one scaling, and the leftover scaling is exactly the determinant. K_1 is the generalization of the determinant to an arbitrary ring: it captures whatever obstruction remains once you allow yourself to row-reduce as much as possible.
Let GL(R) be the infinite general linear group, the union of the groups GL(n, R) under the inclusions A maps to [A, 0; 0, 1]. Inside it sits the subgroup E(R) generated by all elementary matrices. Whitehead's lemma shows E(R) is exactly the commutator subgroup of GL(R), so the quotient K_1(R) = GL(R) / E(R) is abelian. Concretely, K_1(R) is the abelianization of GL(R): the largest abelian quotient of the stable invertible matrices.
For a commutative ring R, the determinant gives a surjection K_1(R) -> R^* (the units), split by viewing a unit as a 1x1 matrix, so K_1(R) = R^* ⊕ SK_1(R), where SK_1(R) is the special part coming from matrices of determinant 1. For fields, Euclidean domains, and local rings SK_1 vanishes and K_1(R) is just the unit group R^*. The nonvanishing of SK_1 is a genuinely subtle, often arithmetic, phenomenon.
For R = Z, the units are {+1, -1} and SK_1(Z) = 0, so K_1(Z) = Z/2Z, detected by the sign of the determinant. For a field F, K_1(F) = F^*, the multiplicative group of nonzero elements.
K_1(Z) = Z/2Z, generated by the matrix of determinant -1.
Whitehead's lemma is the engine here: it states [E(R), E(R)] = E(R) and E(R) = [GL(R), GL(R)], proved by the identity that any commutator of block-diagonal matrices is elementary. This is why K_1 is automatically abelian without imposing abelianization by hand at the level of each GL(n).