Shikai is the initial/first release stage of a Zanpakutō. In this form, the Shinigami gains access to the Znapakutō's true form and powers/abilities. Before achieving Shikai, usually a Shinigami will have a normal sword. They will obtain access to their Shikai after they learn their Zanpakutō's name.
Bankai is a second level of Zanpakutō release. It means "Final Release". By achieving Bankai, Shinigami will have a power up. Example:Tensa Zangetsu (Shikai:Zangetsu) will improve their master's speed and combat. Each Bankai has it's own speciality and limits (including Shikai).