Pole beans are vining versions of the common bean (P vulgaris). They can be developed for use as dried, green shell, or snap beans) Common cultivars are Kentucky Wonder, Blue Lake, Fortex, Missouri Wonder, Romano. They are are available in green, purple and yellow (wax) forms. There are cultivars for just about every region of North America.

Runner beans are usually vining, although there are a few bush cultivars. The are P. coccineus a separate species from common beans. They prefer a cooler climate. For many years used primarily as ornamentals, but they are edible. These have big showy blossoms. Cultivars range from the traditional Scarlet Runner ( scarlet blossoms)to Painted Lady (multicolor blossoms) to Apricot ( apricot color blossoms)to Sunset ( light pink blossoms) to White Lady ( white blossoms) Runner beans are popular in the UK and northern Europe which has an ideal climate for them.

To the excellent summary above, I will add:

Common beans (P. vulgaris) have cotyledons (seed halves) that emerge from the ground during germination. When you see a photo of bean seedlings, with their nodding heads, they are most often P. vulgaris. The flowers tend to be borne in small clusters; and with the exception of a few which are bright purple, they are generally pale & unattractive. The pods are fairly smooth, or slightly hairy.

Runner beans (P. coccineus) have hypogeal seedlings; the seed halves remain underground, and only the leaves emerge. Runner bean seeds also tend to be larger, and usually have shiny seed coats; the predominant colors (in various combinations) are purple, black, white, and brown. The large decorative flowers are borne in long attractive racemes, which continue to bloom over an extended period. The pods are usually rough-textured.

You also need to consider the possibility that an unknown bean is a lima (P. lunatus). Limas are usually flattened; but since many common beans have similar shapes (such as Romano types) this is not a positive means of identification. Also like common beans, the seed halves emerge during germination. Perhaps the first identifiable sign of a lima is the foliage, which is smooth & slightly shiny. Like runner beans, the flowers are borne in long racemes; but they tend to be small & inconspicuous. The pods are short & contain few seeds, typically 3-5.

If you receive bean seeds & are uncertain of the species, these are some of your best clues.