User summary, based on info from Japanese Wikipedia:
Enix, founded in 1975, was one of Japan's largest video game software publishers and book publishers. They were merged with Square in 2003, forming the company Square-Enix.
In September 2, 1975, Yasuhiro Fukushima founded a service center to publish informational magazines for housing, and founded Enix as a child company for software publishing. The name "Enix" comes from Eniac, the computer said to be the first digital computer in the world, and the immortal bird "Phoenix".
In 1982, they started the first Game Hobby Program Contest in Japan. The first price was a million yen. Many of today's game creators were discovered during this contest, Yuji Horii of later Dragon Quest fame being one of them. It was after this contest that Enix entered the Famicon market to create Dragon Quest. Interesting enough, information about the Dragon Quest series was first monopolized by Shueisha, and not in Enix' own magazines. Due to this, people said that the game publishing part of Enix and the book/magazine part of Enix were on bad terms, and it also happened that comics Enix was publishing were made into games by rivalling companies instead of the Enix game publishing team.
Enix was famous for not focusing on in-house programmers, but instead out-sourcing the developpment of games to others. Even in the current in-house programmer based Square-Enix, the game series influenced by the old Enix style are still being developped my outside developpers. In the old Enix era, of Enix' mostly used developpers one finds Chunsoft, Tri-ace and Quintet.
In the comic industry, Enix is mostly famous for their game-style comic magazines Gangan and G-Fantasy.
The 1st of April 2003 they merged with Square and became Square-Enix.