An anime composer is a composer who mainly composes music for anime productions.
There have been many anime composers over the years, and while anime soundtracks are big business in Japan, there have been few notable, long-term composers of anime music before the 2000s.
Joe Hisaishi, best known for his collaboration with Hayao Miyazaki beginning in the mid-1980s. Since most of Hisaishi's anime music has been for Miyazaki, his influence has been somewhat muted compared to later composers.
Shigeaki Saegusa, composer for Mobile Suit Zeta Gundam in 1985, was a classical composer who produced a symphonic score for this series, which went on to be extremely popular (one of the foundation successes of the Gundam franchise). While Saegusa produced only a little more anime music, his Zeta Gundam soundtrack is still considered a classic among otaku. For many of them, Saegusa and Hisaishi were the first to inspire the idea that anime music could be of very high quality.
Kenji Kawai was producing scores for series such as Blue Seed, Patlabor, and Ranma 1/2. While few of these scores were groundbreaking, they were almost all solid works of music. Kawai was arguably the first composer to produce a number of anime soundtracks and achieve at least a modicum of popularity within the otaku community while doing so.
Yoko Kanno garnered some interest with her soundtracks for Escaflowne and Macross Plus during the 1990s, but it was her soundtrack for Cowboy Bebop in 1998 that made her extremely popular among anime fans.
Taku Iwasaki (the Rurouni Kenshin OVAs, Witch Hunter Robin, Read or Die TV, Soul Eater) and Yuki Kajiura (Noir, .hack//SIGN, Kara no Kyoukai) have both produced several well-respected soundtracks in the late 1990s and 2000s.