Like many animals, the elephant was named at a time when the science of taxonomy was in its infancy and the relations between animal groups was just becoming clear. The word elephant is of ancient origin, arriving in English around 1300 via the Old French olyphant, which was reintroduced around 1550 on the Latin model from elephantus from the Ancient Greek genitive of the word elephas, elephantos (ἐλέφαντος) that was used to mean both elephant and ivory. When German zoologist and entomologist Johann Karl Wilhelm Illiger (19 November 1775 – 10 May 1813) named the order to which elephants belonged, he naturally turned to their most distinctive feature, their incredible long prehensile noses, and named the order Proboscidea, from the Ancient Greek proboskis (προβοσκίς) and the Latin proboscis, meaning nose.
Even the scientific classification system of naming is a little bit off and showing its age: of the two species alive today, the smaller, Asian Elephant is known as Elephas maximus while the larger African elephant is known as Loxodonta elphanta, meaning oblique sided tooth. The earliest proboscid dates back 65 million years to the Paleocene: the Eritherium was not nearly as large as modern elephants, and the next oldest probiscid (dating back 56 million years) was the Phosphatherium about the size of a small fox.
Image of elephants courtesy William Warby, used with permission. Check out his cool flickr page.