The New Age school of music emerged (at least in terms of the name and public awareness) in the 1980s. However, it's a safe argument that, depending on each person's definition and parameters of what is and is not New Age music, there were artists creating music that would fit into the category before the 1980s. New Age gained negative connotations even before the decade ended, leading to a decline in the amount of music being labeled as such.
The goal of the genre was to create a calm and peaceful sanctuary of sound. In many ways, New Age music was tied to the New Age movement's interest in meditation and other forms of inner reflection. Beyond that point, it's hard to generalize about New Age music. Some artists produced sounds that were somewhat staid and monolithic, while others produced sound tapestries that were rich and varied. Both acoustic and electronic examples exist, and the music can sit near rock, jazz, folk or classical.
There are a number of types of music that fall within the heading of New Age. Some of these are tied to, and essentially are spin-offs from, other styles of music. Neo-classical is one of them. Electronic progressive rock is another. Chamber jazz, space (sort of a spin-off from space rock) and Celtic New Age are a couple others. Although there is still music being released as New Age today, much of what was called New Age music in the 1980s would now be referred to as "ambient."
Certain musicians are tied to the New Age style of music, with most (or all) of their musical output fitting firmly inside the boundaries. Some of them aren't as well known, but there are many who reached a level of fame. Enya (and her group Clannad) would be included in that group. Vangelis, Enigma, Yanni and Jim Brickman all represent different angles of New Age music. Mannheim Steamroller, best known for its holiday music, fits into the New Age category.
Several musicians did at least one or two New Age albums despite starting in (and being better known for) another genre of music. Jon Anderson, Rick Wakeman and Steve Howe, all alumni of the progressive rock group Yes, fit into that group. Musicians best known for their work in jazz (and progressive rock) who also delved into New Age music include Larry Fast, Jean Michel Jarre and Steve Hillage.