There are six major groups in Euphorbia: sect. Anisophyllum, sect. Crepidaria, sect. Alectoroctonum, sect. Poinsettia, subg. Esula, and sect. Nummulariopsis.
Section Alectoroctonum (synonymous with the genera Agaloma and Tithymalopsis) is a diverse group and is represented by herbaceous and succulent plants with primarily alternate leaves that are symmetric at the base and entire. Many members in this group have petal-like appendages, which differentiate it from sect. Numulariopsis and subgenus Esula. The cyathia are generally not held in a single terminal pleiochasium, as is usually the case in sect. Poinsettia (E. marginata and E. corollata are exceptions to this). It includes plants like Candelilla (Euphorbia antisiphylitica), Snow on the Prairie (Euphorbia marginata), Flowering Spurge (Euphorbia corollata), Cliff Spurge (Euphorbia misera), and many others.
Section Anisophyllum (synonymous with genus Chamaesyce) represents the group of herbaceous plants with opposite leaves with unequal bases, dorsi-ventral stems, interpetiolar stipules, and C4 photosyntheis. The group includes the mat-forming plants like Spotted Spurge (Euphorbia maculata), Prostrate Spurge (Euphorbia prostrata), Whitemargin Sandmat (Euphorbia albomarginata), and many others. For a full list of species in the US, go here: http://www.inaturalist.org/lists/459735-Euphorbia-of-the-United-Statess-Check-List?q=&taxon=506796&observed=any&rank=species&taxonomic_status=active
Section Crepidaria (synonymous with genus Pedilanthus) represents a group of succulents (similar to Candelilla) with zygomorphic (bilaterally symmetric) cyathia. In the US, it is only known from cultivation and potentially a small population in southern Florida. This group includes Slipper Plant (Euphorbia lomelii) and Devil's Backbone (Euphorbia tithymaloides).
Section Poinsettia (synonymous with genus, well, Poinsettia) represents a group of herbaceous plants with large bract-like leaves that are often colorful. The cyathia usually have 3 or fewer glands per cyathium but a couple species have more. The cyathia are also held in inflorescences that terminate the apices of stems. The group includes Poinsettia (Euphorbia pulcherrima), Green Poinsettia (Euphorbia dentata), Painted Leaf (Euphorbia cyathophora), and others.
Section Nummulariopsis represents a small group of erect herbaceous plants in the Southeastern US that appear superficially similar to the narrow-leafed species in sect. Alectoroctonum or like a small, linear-leafed species of subg. Esula. The group includes Euphorbia inundata, E. floridana, E. rosescens, and E. telephioides. For more information about the US species, please click here.
Subgenus Esula (synonymous with genus Tithymalus) represents a large group of mostly herbaceous plants that typically have terminal pleiochasia (inflorescences), alternate stem leaves, and opposite inflorescence leaves. Widespread and represented by many species. The group includes plants like Petty Spurge (Euphorbia peplus; not to be confused with E. peplis), Cypress Spurge (Euphorbia cyparissias), Caper Spurge (Euphorbia lathyris), and Leafy Spurge (Euphorbia esula).
Subgenus Esula sections
Section Chylogala (I) - Section of single species introduced to the United States. Only representative in the US is Euphorbia serrata and it may have been eradicated.
Section Esula (I) - Section of only species introduced to the United States.
Section Exiguae (I) - Section of only species introduced to the United States.
Section Helioscopia (N/I) - Section of species both native and introduced to the United States. US species characterized by glands that lack horns, leaves that are serrated, and fruits that are usually warty.
Section Lathyris (I; monotypic section) - Section of single species introduced to the United States. Lower leaves opposite, often forming an X-shape when viewed from the top. Gland horns flattened and rounded.
Section Myrsiniteae (I) - Section of only species introduced to the United States. Leaves thick, somewhat succulent; gland horns flattened, broadest towards the tips.
Section Pachycladae (I) - Section of only species introduced to the United States. Two very dissimilar species occur in this section: E. dendroides and E. terracina. E. dendroides is woody and has glands without horns. E. terracina has serrated leaves and glands with horns.
Section Paralias (N/I) - Section of species both native and introduced to the United States. Mostly shore species with closely set leaves but includes the introduced E. segitallis which is currently only known in the US from San Diego California. Also includes E. paralias and E. trichotoma.
Section Patellares (I) - Section of only species introduced to the United States; only represented by E. amygdaloides and E. characias. Inflorescences large and showy.
Section Pithyusa (I) - Section of only species introduced to the United States.
Section Tithymalus (N/I) - Section of mostly species native to the United States; only E. peplus is introduced.
Some commonly cultivated groups
Subgenus Euphorbia
Section Cubanthus
Section Euphorbia
Section Goniostema
Section Monodenium
Section Tirucalli
Subgenus Athymalus
Section Anthacanthae
Total list of sections (64):
Subgenus Euphorbia (21)
Section Brasilienses
Section Calyculatae
Section Crepidaria
Section Cubanthus
Section Denisophorbia
Section Deuterocalli
Section Euphorbia
Section Euphorbiastrum
Section Goniostema
Section Lactifluae
Section Mesophyllae
Section Monadenium
Section Nummulariopsis
Section Pachysanthae
Section Pacificae
Section Pervilleanae
Section Portulacastrum
Section Rubellae
Section Stachydium
Section Tanquahuete
Section Tirucalli
Subgenus Athymalus (7)
Section Anthacanthae
Section Antso
Section Balsamis
Section Crotonoides
Section Lyciopsis
Section Pseudacalypha
Section Somalica
Subgenus Esula (21)
Section Aphyllis
Section Arvales
Section Biumbellatae
Section Calyptratae
Section Chylogala
Section Esula
Section Exiguae
Section Guyonianae
Section Helioscopia
Section Herpetorrhizae
Section Holophyllum
Section Lagascae
Section Lathyris
Section Myrsiniteae
Section Pachycladae
Section Paralias
Section Patellares
Section Pithyusa
Section Sclerocyathium
Section Szovitsiae
Section Tithymalus
Subgenus Chamaesyce (15)
Section Alectoroctonum
Section Anisophyllum
Section Articulofruticosae
Section Bosseriae
Section Cheirolepidium
Section Crossadenia
Section Denisiae
Section Eremophyton
Section Espinosae
Section Frondosae
Section Gueinziae
Section Plagianthae
Section Poinsettia
Section Scatorhizae
Section Tenellae