This exception falls under the Veteran Free Agent exceptions in the Collective Bargaining Agreement. A player qualifying for this exception is known as non-qualifying free agent. They are known as "non-qualifying" because they do not qualify for the Bird or Early-Bird exceptions.
This particular exception allows teams to re-sign a player at 120% of his previous salary or 120% of the NBA minimum, whichever is greater. The maximum length of a Non-Bird Exception contract is four years (the previous CBA set the maximum contract length at 6 years). Like the other Larry Bird exceptions, this exception may only be used by a team to re-sign their own free agents, not another team's free agents. This gives teams a slight advantage in signing their own players, as they are allowed to go over salary cap without penalty. Another interested team may incur luxury tax penalties for signing the player if it puts them over the salary cap.