The difference between "half duplex" and "full duplex" is the difference between a walkie talkie and a phone.
In digital communications, half duplex means data can only go in one direction at a time. That's the way walkie talkies work. One person communicates and then releases the transmission button allowing the other person to talk.
Fully duplex means data can be transmitted and received at the same time. We know from experience that people who talk over the phone can talk simultaneously.
On the technical end, half duplex is synonymous with TDD (time-divisional duplex), which transmits and receives data at different times on the same frequency. It is used frequently in wireless bridging since it doesn't impose the extra cost of hardware and software needed for separate transmit and receive antennas that a full duplex system requires. Home Wi-Fi and Bluetooth use TDD, which can support data as well as voice communications.
Full duplex is synonymous with FDD (frequency-division duplex), which transmits and receives data using two different frequencies. It's more expensive since, as mentioned before, the transmit and receive antennas require their own hardware and software systems. You'll typically find these types of solutions used in commercial and enterprise class situations.