Youden square is a reduction from randomized complete blocks to a balanced incomplete design in which one row or column has been removed from a Latin square (so it is no longer square). It is a useful design for balancing out the effects of treatment order in a repeated-measures sequence. The example on these web pages could pertain to a predator odour treatment (A with a = 4 levels) tainting the food offered to each of a = 4 mice (random factor B). Each mouse is tested with a-1 = 3 odour types in order (random factor C) assigned by the Youden square.
Doncaster, C. P. & Davey, A. J. H. (2007) Analysis of Variance and Covariance: How to Choose and Construct Models for the Life Sciences. Cambridge: Cambridge University Press.