Asynchrony, in the general meaning, is the state of not being synchronized.
At the higher data link layer of communication, asynchrony is synonym of statistical multiplexing, such as in packet mode. The information transmission may or may not start immediately as requested by sender, the additional delay being caused by medium congestion. Contrast with example of circuit switched communication, which (once circuit is established) allows immediate start of transfer with a guaranteed bit rate. Confusingly, a communication is often synchronous at the physical layer, while being asynchronous at the data link layer.
In programming, asynchronous events are those occurring independently of the main program flow. Asynchronous actions are actions executed in a non-blocking scheme, allowing the main program flow to continue processing.