The genus mammoths, in latin Mammuthus, was a group of species, belonging to the family of elephants, entirely separated in taxonomy from the Mastodons and the genus family Mammutidae, although they sometimes shared the same envoronment. (For scientific reasons, the mastodons was renamed to family Mammutidae, which became a source for future confusion and misunderstandings). The Mammoths probably has origin from Stegodon and started to develop during upper Pliocene, (some 4 million years ago) in africa, spred to europe and asia, and Mammuthus meridionalis went over Bering Strait to north america about 1.8 million years ago, (the wooly Mammoth went over much later) and became extinct during lower Holocene, probaby exterminated by prehistoric humans who hunted elephants and Mammoths and made huts and houses out of their bones.
Recently made DNA-tests (Extern reference link) gives the indication that genus Mammuthus is a sister genus to elephas, with a gap to Loxodonta. Mammoths lived in different areas, some in the northern arctic areas, but not all of them, and they developed over a long period into different forms, so when speaking about Mammoths, it should be clear that this is a group of species, and not only the species wolly Mammoth. There were several recognized species: