I attempted to find the answer to this age old question with some online sleuthing.
The exact definition is a bit sketchy and different sources have different explanations, but this is what I came up with...
Mandarin orange: A loose-skinned orange category that includes several varieties that can be sweet or tart, seedless or not and can range in size from as small as an egg to as large as a medium grapefruit. They all, however, have skins that slip easily off the fruit. Among the more well-known mandarin-orange family members are Clementine, dancy, Satsuma and tangerine.
"Mandarin" and "tangerine" are two words for the same thing, technically Citrus reticulata Blanco. They're called mandarins because they were thought to be native to China. The name tangerine comes from Tangier, Morocco, the port from which the first tangerines were shipped to Europe and so they were thought to have come from Tangiers. They are in fact native to southeast Asia someplace, and they exported from North Africa, so both origin myths are correct.
There are many varieties of tangerines, most associated with the word are the Dancy, which is a lovely, loose-skinned, comparatively late season, and scrumptiously delicious tangerine. They don't store well either on or off the tree, and they have seeds, which in recent years has made less commercially desirable, but their flavour still defines "tangerine" for many of us. Tangerines generally have a thicker, rougher skin
For many years the Christmas tangerine has been the Satsuma, a loose-skinned seedless early maturing variety which originated in Japan. Most of the canned mandarin oranges on the market are satsumas.
The tiny Clementine has a thin peel and a tangy-sweet red-orange flesh that's usually seedless. It's cultivated in Spain and North Africa. Clementines are a family of tangerine varieties.