The chairman is typically elected by the board, but there's a lot of variety. Some smaller ones just make a schedule early to basically rotate through members for two years periods. I think there's a lot of variety in this part depending on the company, board, shareholders, etc.
The chairman is the head of the board. He doesn't report to the board. He, along with the rest of the directors, reports to the shareholders.
The chairman has a fiduciary duty to the shareholders, not to his fellow board members.