An asynchronous circuit is a circuit in which the parts are largely autonomous. They are not subject to a central clock circuit, but instead need only wait for the components used in the current instruction/operation. This is contrasted in a synchronous circuit in which all parts of the circuit are operational and drawing power, even if not used for the current instruction/operation.
Asynchronous circuits offer several benefits over synchronous circuits, including avoiding clock skew and being (typically) less noisy.