The Amnesty Clause is a clause negotiated into the newly ratified Collective Bargaining Agreement between the players and owners which allows a team to release one player from their roster and be free of any financial obligations to that player. This was added to allow teams to clear salary cap space in preparation for new salary cap rules. A player who is released under this program is said to be "amnestied."
This is the second time an NBA CBA has included an Amnesty Clause. Each team is only allowed to amnesty one player for the entire period of the CBA, and must select a player to be amnestied before the start of the season. There is belief among basketball experts that such a rule is put in place by the owners to rid themselves of bad contracts that they had previously signed. The original rule was sometimes called the "Allen Houston" because the New York Knicks had signed Allen Houston to a $100M contract that was untenable based on his performance.