Page "Commutator subgroup" Paragraph 1

The commutator subgroup is important because it is the smallest normal subgroup such that the quotient group of the original group by this subgroup is abelian.

In other words, G / N is abelian if and only if N contains the commutator subgroup.

So in some sense it provides a measure of how far the group is from being abelian ; the larger the commutator subgroup is, the " less abelian " the group is.