A cheap very old intercom is half-duplex. I press a button to speak and you hear it. But while I am pressing the button it disconnects your microphone so I can't hear you.
If you speak then you press your button that disconnects my microphone. So you cannot hear me.
Full-duplex allows each person at each end to always hear the other end but is a little more complicated. It can cancel its own signal or it can use a modern digital echo canceller circuit.