The name given to a set of pipes producing a given tone (flute, string, diapason, clarinet, etc.). The number of pipes depends on the compass of the stop(s) associated with the rank (eg 8′ Flute and 4′ Flute). Herein lies one of the differences between a classical (‘straight’) and a theatre organ.
For example, a diapason stop on a straight organ will have 61 pipes, all producing the same tone colour but each of a different pitch to cover the compass of the stop from bottom C to top C. Usually, only one stop is associated with a particular rank of pipes.
A theatre organ rank differs slightly because the theatre organ is based on the unit/extension principle. This means that the rank may be played by stops at different pitches so if, for example, the Diapason is available at 8′ and 4′ pitches, then there will be 73 pipes (61 plus another 12 for the additional 4′ octave). The most extended rank on theatre organs is usually the Flute which can be available from 16′ to 2′, in which case there will be 61+12+12+12 = 97 pipes in the rank.
The section on Extension and Unification explains this further.