Half-duplex is used to describe communication where only... one side can talk at a time. Once one side has finished transmitting its data, the other side can respond. Only one node can talk at a time. If both try to talk at the same time, a collision will occur on the network.
As you can understand, this method of communication is not very efficient and requires more time to send/receive larger amounts of data.
Older networks used to work in half-duplex mode, due to the constraints of the network medium (coax cable) and hardware equipment (hubs).
On the other hand, full-duplex is used to describe communication where both sides are able to send and receive data at the same time. In these cases, there is no danger of a collision and therefore the transfer of data is completed much faster.
Today, all networks make use of switches (rather than hubs) and UTP Ethernet cabling, which allow full-duplex communication between all connected hosts.