In Gorenstein's Finite Group Theory book on page 189,
"It will be convenient to introduce the terms depth, normal depth, and characteristic depth of a $p$-group $P$ to denote the maximum of $m(A)$, where $A$ ranges respectively over all abelian, all normal abelian, and all characteristic abelian subgroups of $P$."
I could not really understand the definition of depth, normal depth, and characteristic depth from above part. If someone could explain these terms, I would be thankful.