Euphorbia species of the United States's Journal

I've heard it mentioned a couple of times that it helps a lot for people to have an idea of what they need to photograph. Hopefully, this will help.

1. Habit.
This is tied for the most important thing to photograph. If the photo is clear, all the essential characteristics will be visible in one or two good habit shots. In plants that lie flat along the ground, this only needs to be a top-down photo. In something that is upright, it's important to get a top-down photo and one from the side or potentially one at an angle in-between.

2. Branches.
Stems with leaves, flowers, and fruit if present. This one is just as important as habit. This allows for seeing a lot of major characteristics including leaves, phyllotaxy, stipules, cyathia, and sometimes fruit.

3. Undersides of leaves and/or branches (generally optional).

This helps with some groups and may help one see styles which are important in a few species. This can be particularly helpful for the species that lie flat along the ground.

Close-ups of the characteristics themselves (all usually optional):

Stems.
For some species, you may need information about the hairs on the stems or even the stems themselves. To facilitate this, a close-up of the stem can be helpful. For instance, one useful way to distinguish E. hyssopifolia from E. nutans is be looking for internode hairs (long and straight or absent in the former; short and curly in the latter). Stems can also be winged (E. serpillifolia), somewhat flattened or completely round.
Species where this level of detail may be helpful: E. hyssopifolia, E. nutans, E. dentata, E. davidii, E. serpillifolia, and E. glyptosperma.

Leaves.
Depending on what group you photograph, there may be more than one type of leaf. This is particularly important in subgenus Esula, which have the typical stem leaves and then two types of bracts. Many members outside subgenus Esula have leaves that are different when growing under a cyathium. When this is the case, it can be useful to photograph all different types. It's always best to get photos of both sides of the leaves, though usually not necessary in the US members of the genus Euphorbia.
Species where this level of detail may be helpful: members of subgenus Esula, E. bicolor, E. marginata, possibly E. dentata, and possibly E. davidii.

Stipules.
Small triangular (often membraneous) structures between two pairs of leaves in section Anisophyllum. Stipules are technically made from leaf tissue at the base of the petiole. Stipules vary from being entire to deeply lobed or even fused with its neighboring stipule. These may be hairy or non-hairy. In some, they are even reduced to simple glands.
Species where this level of detail may be helpful: members of section Anisophyllum.

Cyathia.
Morphology explained simply here. Diversity in cyathial form within Euphorbia is vast. Though not always necessary for an ID, the uniqueness of these structures make them fun to photograph. Nearly every aspect of the cyathium may differ between species, but the most readily visible and often most useful diversity can be found in the glands and their appendages. The number of glands can vary from 1 to 5, and typically not 2 or 3 (rarely, plants will produce 2, 3, 6 and very rarely more glands). The gland ornaments can be petal-like, horn-like, or absent. If petal-like, they can be divided or undivided. They are usually white but can be pink, red, or (if horn-like) even yellow. These characteristics are usually visible from the branch photos but are fun to look at more closely.
Species where this level of detail may be helpful: essentially all, though especially E. arizonica, E. setiloba, E. spathulata, E. texana, E. ocellata and look-alikes, E. missurica, E. parryi, and eastern species of section Alectoroctonum.

Fruits and styles.
The fruits are not quite as variable as cyathia but are no less important. Fruits may be hairy or not hairy; warty, papillose, or smooth; and/or deeply lobed or subtriangular in cross-section. The three styles found at the apex of the fruit and maturing ovaries usually have two branches which may be divided all the way to the base, not divided at all, or be divided somewhere in-between.
Species where this level of detail may be helpful: essentially all.

Seeds.
Though difficult to photograph and only completely essential to identification in a couple of species pairs in the US, the seeds can be really important to identification. If you are unfamiliar with the group and have to key them out, you essentially must look at seeds. Seeds may be smooth, ridged, papillose, pitted, carunculate, ecarunculate, quadrangular, or round. Tips for extracting and photographing seeds can be found here.
Species where this level of detail may be helpful: essentially all, though especially E. simulans, E. theriaca, E. hyssopifolia, E. nutans, E. serpillifolia, E. glyptosperma, E. dentata (preferably with scale), E. davidii (preferably with scale), E. pubentissima (with scale), and E. corollata (with scale).

Seed morphology is very important to the taxonomy of the genus Euphorbia. It is so important that most larger keys require looking at the seeds. The seeds are so diagnostic that there are many instances where species can be determined just by looking at the seeds alone. Because of this, I think it's time that I write a little bit about the more complicated aspects of Euphorbia seed morphology. This does not cover the full variability of seed morphology but is intended to give an introduction to help you understand the terminology.

In almost every Euphorbia seed, there are four more or less flat sides (referred to here as faces). Two of these are generally longer than the other two as shown below.

These are useful for identification, but in order to understand the morphology of the seeds, there are some other characteristics that should be taken into account. First off, the faces are best identified in their relationship to the four angles. Look at the figures below.

Left and Right: A: distal angle (angle furthest from center of fruit); B: lateral angles; C: proximal angle (angle nearest center of fruit; also called the raphe). Left only: D: testa (seed coat); E: mucilaginous coat (only the white part). Right only: D: chalazal end; E: hilum (micropyle visible as brown dot directly right of hilum).

The two faces adjacent to the distal angle are known as distal faces. The two angles adjacent to the proximal angle are known as proximal faces. There is considerable variation in these faces between species. In some, the faces bulge so much or the angles compress so much that the faces are nearly indistinguishable from the angles and the seeds become essentially round in cross-section. In some, these faces are compressed and create somewhat deep depressions between the angles. The topography of the angles varies too. They may be ridged, wrinkled, pitted, papillate, or have some variation of these. In some species, it is important to know whether the ridges overlap the angles. Some have sharper ridges than others.

Aside from faces and angles, the figures above also identify the testa (the seed coat), the mucilaginous coat (the part that makes the coat slimy/sticky when wet), the chalazal end (defined as the end of the raphe opposite to the hilum), the hilum (place where the seed was attached to the fruit), and the micropyle (place where the pollen entered into the ovule during fertilization). Many of these exhibit variability between species that make them diagnostic. A characteristic not shown in the above figure is a caruncle (an outgrowth of the hilum), which can be useful for identification.

Examples:

Euphorbia carunculata with an outgrowth of the micropyle. Source observation.

Euphorbia lata showing connections between hilum, micropyle, and fruit tissue. Source observation.

Euphorbia alta seed with caruncle. Source observation.

Euphorbia roemeriana displaying caruncle and enlarged chalaza. Source observation.

Seeds of various species showing diversity in ridges. Top left: E. serrula; top right: E. simulans; bottom left: E. prostrata; bottom center: E. glyptosperma; bottom right: Euphorbia theriaca var. theriaca.