English, Scottish and Irish, derived from the Gaelic name Niall. The meaning may be "cloud", passionate", or "champion". Whyte favours the last meaning but ONC only gives it as 'possible'.
Niall is probably derived from the Norse name Njal.
The Latin form in the Middle Ages was Nigellus, from "black", which also gave rise to Nigel. We have therefore classified the relationship between Neil and Nigel as Lesser Synonyms.
A correspondent [MG2] has questioned whether the Scandanavian names Nils, Neils are equivalent to Neil but it is fairly certain they are variants of Nicholas. That is not to say that they might not be Anglicized as Neil in individual cases.
A correspondent [BW] has noted that Neil/Neily may be used as pet names for Cornelius.