When peonies are offered for sale, they are sometimes classified by their botanical type. The categories most often seen in nursery catalogues are: (a) species, (b) herbaceous, (c) tree peony, and (d) intersectional. The following tries to clear up what these terms generally mean.
Species
There are somewhere between 35 and 50 different species of peony (i.e. plants belonging to the genus Paeonia). However, most of these botanical species grow wild in Asia, southern Europe, or western North America. Very few of them are culivated and sold commercially. The few that are available are usually not as large and showy as the peonies we see growing in neighbourhood gardens.
When the term "species peony" is used by a nursery or commercial grower, it basically means: "wild peony which we've 'captured' and made available to you non-explorers because you want something rare and different". While they are rare, and different from most garden peonies, they are also not as spectacular bloomers.
The ones most often available commercially are: Paeonia anomala subsp. veitchii (formerly known as Paeonia veitchii), Paeonia daurica subsp. mlokosewitschii (formerly known as Paeonia mlokosewitschii, Paeonia officinalis, Paeonia officinalis mollis (formerly known as Paeonia mollis), Paeonia peregrina, and Paeonia tenuifolia. They are described in more detail in my section on Species Peonies.
Herbaceous peony
The species of peony which is probably most important commercially is the Paeonia lactiflora. This species is native to central and eastern Asia. In particular, it occurs in northern China, and has been cultivated there for perhaps 1,600 years.
There are at least 3,000 registered cultivars (i.e. cultivated varieties) of Paeonia lactiflora (and probably even more which have never been registered or marketed). Most of the peonies sold commercially are either culivars of Paeonia lactiflora, or else hybrids, with some of its genes.
Tree Peony
Tree peonies (Paeonia rockii) have woody stems that lose their leaves in the fall, but the woody stems stay intact. They tend to bloom earlier and with larger flowers than the herbaceous peony. Despite this, they are generally not as popular as herbaceous peonies, because they are such slow growers. They have been of most commercial interest recently as a source of hybrids, which are called "Intersectional peonies".
Intersectional Peony
Intersectional Peonies are hybrids; they are derived from a cross between a white Herbaceous Peony (Paeonia lactiflora 'Kakoden') and a yellow hybrid Tree Peony (Paeonia x lemoinei) carried out in 1948 by Mr. Toichi Itoh. For this reason, they are also known as Itoh hybrids or Itoh peonies.
Intersectionals are valued because they are available in colours that traditional peonies can't produce — in particular, more intense shades of yellow, peach, and coral. These plants have the lovely leaf form of the tree peonies, but die to the ground in the winter like herbaceous peonies. The plants are strong and healthy with a nice rounded bush form, but are generally shorter than most bush peonies. Since they are recent introductions and are still in short supply, they are usually quite expensive.
Examples of some intersectional peony varieties are: Bartzella, Border Charm, Cora Louise, First Arrival, Garden Treasure, Hillary, Julia Rose, Sequestered Sunshine.