Bankai is a release style for a soul reaper zanpakuto. This mode is relaesed after the shikai is at its full strenght, and can be used to increase the speed, endurance, attack or even give the weilder special attacks.
Bankai has only been known to appear in the zanpakuto of the most seasoned of noble soul reapers, and even then it takes for ever for them to develop a full strength and fully develloped bankai. The first soul reaper to actually devellop a bankai and not be a noble or a captain is ichigo Kurosaki. the second non noble was an assistant captain and that was Renji Abarai. He is Byakuya's assistant.
There is one last stage after bankai and that is hollowfication. That iss when a soul reaper breaks the barrier between the hollow and soul reaper worlds.