While “rank” has several meanings in the “real world,” such as defining the level a person might attain in the armed services (sergeant, major, and so forth), in the world of music it refers to the number of pipe groups in a pipe organ or the sets of strings in a harpsichord.
For example, the famous Westminster Cathedral organ (c.950) had a compass of 40 notes with ten ranks of pipes. The organ was so huge that it required two people to play it and no less than 70 strong men to work the 26 bellows. Smaller organs might have as few as two or four ranks. Each rank would typically include groups of pipes designed to produce a distinctive sound, not too different from our modern preset banks on a synth if you think about it. Ranks might include “stops” that would produce flue (not flute) or reed tones with such descriptive (and sometime whimsical) names as “Viola da Gamba,” “Gemshorn,” “Cornet,” and “Trompette.”
On a harpsichord, “ranks” indicate sets of strings, so that a small instrument might have just one rank of strings pitched at 8′ while a double rank harpsichord would have two ranks (or sets), usually voiced at 4′ and 8′.