Very Early Season – mid spring – April
Early Season – late spring – May
Mid Season – early summer – June
Late Season – mid-summer – late June-early July

Paeoniaceae – the family into which all peonies fall.

Peonia – the Genus name

lactiflora, officinalis, peregrina etc. – the Species name or name of a wild peony

‘Festiva Maxima’, ‘Bowl of Beauty’, ‘Red Charm’ – these names in single quotes are those of peonies propagated by vegetative means to always produce the same plant, and are recognized as such under the rules of the International Code of Nomenclature for Cultivated Plants.

(Saunders, 1942), (Dessert, 1908), (Menard, 2004) – these indicate the breeder, and the date of introduction

So...

Paeoniav eitchii, or Peoniam ascula – indicates a species or true wild plant. One must be careful to purchase these from reliable sources as there are many
open pollinated seedlings on the market, which are most certainly hybrids.

And...

‘Le Printemps’ (Lemoine, 1905) – refers to a plant known as Le printemps which was named in 1905 by the breeder Lemoine, and which is a cultivar or a hybrid

A cultivar (a cultivated variety) is a plant bred or chosen to have certain desirable traits.

A hybrid refers to a cultivated plant containing the genes of two or more different species (eg. Paeonia lactiflora X Paeonia officinalis). A quadruple hybrid for example contains genes from four different species.