To quote Wikipedia:"DNA testing has now shown that there are in fact three extant elephant species: the two African types (formerly considered to be separate populations of a single species, the African Elephant) and the South Asian species, known as the Indian or Asian Elephant. The North African elephant of Hannibal fame was a now-extinct fourth species or a subspecies of the Forest Elephant (Loxodonta (africana) pharaoensis); it disappeared around the 1st or 2nd century CE. ...Differences include the African Forest Elephant's long, narrow mandible (the African Bush Elephant's is short and wide), its rounded ears (an African Bush Elephant's ears are more pointed), different tusks, and considerably smaller size. The Male African Forest Elephant rarely exceed 2.5 metres (8 feet) in height, while the African Bush Elephant is usually over 3 metres (just under 10 feet) and sometimes almost 4 metres (13 feet) tall. ...Asian Elephants have other differences from their African relatives, including a more arched back than the African, one semi-prehensile "finger" at the tip of their trunk as opposed to two, 4 nails on each hind foot instead of three, and 19 pairs of ribs instead of 21. Also, unlike female African Elephants, female Asian Elephants lack tusks. The forehead has two hemispherical bulges unlike the flat front of the African. Some males may also lack tusks and they are termed as makhnas. The population in Sri Lanka has a greater number of makhnas."
There is also some mention in primary sources of a "Syrian" elephant, also now extinct; there is some doubt from the descriptions whether it was a transplanted population of Indians, or a species related to the N. African, or a hybrid.