What are compact cranks?
They probably weren't talking about the cranks per-se, but the whole crankset, which is the cranks and the chainrings. 

Compacts are cranksets with smaller chainrings (typically 50 teeth and 34 teeth) as opposed to standard cranksets which have say 52:42. Combined with wide-range cassette this can give a good range of gears without having to have a triple-ring.  

Biopace were oval rings - you can get non-round chainrings for either compact or regular cranksets.

Note that the cranks themselves do come in different lengths to account for different leg-lengths, but this is independent of the chainring diameter.