Probably a good source is Sullivan’s MIT notes.
LNM304 (B-K) says on p 7 that nilpotent means that the up to homotopy, the Postnikov tower can be refined to a tower of principal fibrations.
Ref: Jardine-Goerss section VI.6. Nilpotent spaces admit a yoga of crawling up a Postnikov tower.
Hilton et al: Localization of nilpotent groups and spaces (Homotopy th folder)
nLab page on Nilpotent space