I have heard derived length, for example, described as a measure of "how non-commutative" the group is. An abelian group will have derived length $1$, whereas a non-solvable group will be so non-commutative that the even the fastest converging commutator series will continue infinitely.
Analogously, nilpotency class measures "how close to being abelian" a group is by finding the length of the slowest converging commutator series. A group of nilpotency class $2$ has essentially only one "twist" in its structure, and a group sufficiently far from being abelian cannot be completely unraveled by commutators without using bigger guns.
Then, between these two definitions, we've got Fitting length, which measures how far a group is from being nilpotent.
Of course, there is another series floating around out there.
The Frattini subgroup of a group $G$, denoted $\Phi(G)$, is the intersection of all maximal subgroups of $G$. (Equivalently, $\Phi(G)$ is the set of all non-generators of $G$.) We define the Frattini series by $\Phi_0(G)=G$ and $\Phi_{k+1}(G)=\Phi(\Phi_k(G))$. The smallest $n$ for which $\Phi_n(G)=1$ is the Frattini length of $G$.
So, intuitively, what does Frattini length measure?