Duplex
If a particular two-way communications channel can carry signals in both directions simultaneously, it is referred to as a full-duplex channel. If such a channel can only carry a signal in one direction at a time, it is referred to as a half-duplex channel. (Here, a "channel" refers to the communications equipment and infrastructure linking two users; in radio communications, this includes frequencies, radios, and repeaters, among other things.)
For example, traditional two-way radio channels are half-duplex, and calls over the telephone network and the Internet are full-duplex. Because of the difficulty of transmitting and receiving radio signals simultaneously, most amateur radio communications is half-duplex, though there are some exceptions. For this reason, special care must be taken to ensure that only one person talks at once and that people who wish to interrupt a contact in progress can do so. This is accomplished through such measures as saying "over" at the end of a transmission while on simplex, repeater time-out timers (where the repeater will stop transmitting after a transmission exceeds a given length, usually three minutes) and courtesy tones (where the repeater transmits a tone after the end of a transmission and a delay to indicate that the transmission has ended and to give others time to interrupt).
Several measures exist to enable full-duplex communications to be sent over radio, which is fundamentally a half-duplex medium (since only one signal can be sent on one frequency at one time). Frequency-division duplexing involves using separate transmit and receive frequencies, and is used by HF stations in pileups, FM repeaters on VHF and UHF. Using separate transmit and receive frequencies is often called split (especially in the context of HF operation) or duplex operation, and it has many applications beyond full-duplex communications. Time-division duplexing involves using a single frequency channel; the two stations using it each alternate between transmitting and receiving fast enough that information can be transmitted bidirectionally without a significant delay. This is used by digipeters on packet and APRS; by relays on JS8; and in QSK operation on CW.
Repeaters
In order to simultaneously receive and trandmit a signal, repeaters must operate as full-duplex devices. This requires special equipment, as the high power of a radio transmitter would overload or damage a receiver if the two were connected directly and operated simultaneously. Several arrangements exist to prevent this. For most VHF/UHF FM repeaters, cavity filters that provide extreme attenuation in a narrow frequency range are used to filter out the signal fed to the receiver; on these frequencies, cavity filters are large but not impractically large, and are somewhat expensive but within the budget of a repeater operator.
Digipeters, as used by APRS, packet radio, and JS8 (though called "relays"), avoid this problem by only starting to retransmit packets after they are fully received. This additionally allows for the use of only one frequency for both digipeter input and output.
QSK operation
Because a CW (Morse code) signal is sent by keying the transmitter on and off, it is possible to achieve something similar to full-duplex operation by rapidly switching between transmission and reception in the interval between the dits and dahs of a CW signal. This is called full break-in or QSK operation, and allows stations to interrupt the station currently sending CW.