Pretty easy to explain: a duplex system is a communications system involving two devices that talk to each other. A half-duplex system means only one side can transmit at a time. The canonical example are walkie talkies. Because walkie talkies operate on a single shared frequency, when you push in the button to talk, you can't hear the other person. (For those of you who have never used walkie talkies, imagine a two-way street that briefly condenses down to a one lane bridge. While traffic from one direction crosses the bridge, traffic from the other side must wait. When the bridge crossers clear, the waiting traffic can now cross.) Contrast this with a full-duplex system, where both sides can communicate with each other simultaneously (like a phone conversation.)
In the early days of wireless and Internet technology, it was quite common to see half-duplex protocols put in place in systems over long distances, since it was much cheaper. Intercontinental telegraph cables, Ethernet lines, and coaxial cable systems were often half-duplex. In addition, early modems were half-duplex in order to eliminate interference (accidental or deliberate) from affecting a transfer. As switches and security were strengthened, modems evolved into the full-duplex ones found today.
Half-duplex is also sometimes referred to as simplex communication, although that word is more commonly used to describe a system in which communication can only travel in one direction at all.