Asynchronous is a term which means that some event, or device, does not operate with a clock, or on fixed timing.
Contemporary logic design is usually synchronous, where a clock signal is used to control when everything changes, but older logic systems were often asynchronous, without use of a clock.
When an incoming signal may change value at any time, relative to a clocked circuit, that signal is said to be asynchronous with respect to that circuit. An synchronizer must be used to bring the external signal's timing into the clocked circuit.
Asynchronous serial lines do not have a clock to indicate timing of data, and the time between two characters sent on such a line may vary arbitrarily.