Full-Duplex:
connection is usually 3 wires: Tx/Rx/GND
Just as Sakbal said: Each device can transmit bytes at any time. For example the RS232 port of the PC works in this way.
Async: Means there is no clock signal transmission for clocking the data. Both connected devices should use the same baudrate (2400,4800,9600 etc) to be able to communicate. The receiving device then will start clocking the data as soon the startbit arrives. Because they use the same baudrate the receiving device knows the bit length, so it knows when to clock the bit in the receive buffer.
The opposite is Sync. Here you need more lines for connection: Tx+Clock / Rx+Clock / GND. The transmitting device will control the clock line. The receiving device will clock the bit at RXline upon change of clock signal at clock line. No need to agree on baudrate or so for the devices.