Synchronous: different circuits are synchronized by clock. The clock cycles could be not only the same single clock but also multiple relationship. For example, 10ns, 5ns, 2.5ns used in three different circuits, which are synchronous.
Synchronous circuit is consist of sequence circuit (register and different flip-flops) and logic circuits. Each process is controlled by strict timing. One single clock is shared by those different circuits and clock rising or falling edge is used to change every status.
Asynchronous: different independent clocks without multiple relationship are used in circuits.
Asynchronous circuit is mainly consist of logic circuits for generating address decoder, FIFO or RAM write/read controlling signal. It can also used in sequence circuit but doesn’t have united clock. The changing status happens at the stable condition. The reliable setup and hold time are necessary in those circuits.