The groups you refer to are considered either subgenera or sections of genus Paphiopedilum, depending on whose classification system you look at. I think the following names would be recognizable to most Paph growers, though not accepted by all taxonomists (some example species follow each name):
Cochlopetalum (sequential flowering species like primulinum and glaucophyllum)
Brachypetalum (bellatulum, niveum, concolor, etc.)
Parvisepalum (delenatii, malipoense,armeniacum, etc.)
Coryopedilum (multifloral species like rothschildianum, sanderianum, philippinense, etc.)
Pardalopetalum (the other multifloral group, like haynaldianum, lowii, parishii)
Barbata (or Sigmatopetalum) (Though 'Maudiae-type' is a purely horticultural designation it basically falls within Barbata, species like callosum, sukhakulii, appletonianum)
Paphiopedilum (the bulldog or 'complex' hybrids come mostly from these species, like gratrixianum, charlesworthii, fairrieanum, hirsutissimum)
Classification traditionally depended on examination of physical characteristics and using those characteristics to separate the 'groups'. Increasingly genetic analysis is used to clarify the relationships. Either way it should reflect the actual evolutionary relationships. You really do need to look at some comprehensive references to learn the technical distinctions, but you can come to recognize them with a little practice without knowing the technicalities. Slipperorchids.info has a good survey of various taxonomic systems with full lists of species, and the Internet Orchids Species Photo Encyclopedia has very good photos for visual comparisons and some botanical details.