What happened
For a decade the armies of Hosokawa Katsumoto (the Eastern camp) and Yamana Sozen (the Western camp) fought entrenched inside Kyoto itself. Both commanders died in 1473; the war petered out in 1477 with no winner, leaving Kyoto largely in ashes.
Background
The war was sparked by succession disputes — in the Ashikaga shogunal house and in the great families.
Consequences
Shogunal authority collapsed, and provincial lords stopped obeying the center. The following century of daimyo warfare — the Sengoku period — saw upstarts overthrow their superiors (gekokujo). The Onin War is often cited as the true end of central rule in premodern Japan.