What are Simplex and Duplex Systems for Two Way Radios?

Simplex
The terms simplex and duplex refer to the communication between two host devices. Simplex has one-way radio control. It offers only one way communication at a time, using a single radio frequency. 
A simplex repeater consists of a radio on a simplex frequency and a digital voice recorder. When a signal is received, the recorder stores the message. When the received signal ends, the digital voice recorder retransmits the message on the same frequency. Operator A sends a message to Operator B. In that process, A's transmitter is on, and his (or her) receiver is off, so B can only listen. When A finishes, typically with the word "over", releasing the push-to-talk switch, then B can send a reply. 
Two operators can communicate in a simplex link only if they can (approximately) see each other. Because the range is short anyway, these transceivers often operate on batteries at low power and can be small and convenient. 
A radio signal is transmitted and it is up to the receiver to correctly determine what message has been sent and whether it arrived intact. Since televisions don't talk back to the satellites (yet), simplex communication works great in broadcast media such as radio, television and public announcement systems.
Duplex
Duplex describes a two-way communication. Both users can talk at the same time (as with cell phones). With most electrical, fiber optic, two-way radio and satellite links, this is usually achieved with more than one physical connection. It is like a two lane bridge on a two-lane highway. In the world of data communications, full duplex allows both way communications simultaneously.
Most of the frequencies on your VHF are "duplex". That means they receive and transmit on different frequencies even though they are on the same channel. This means that even if you are standing next to another radio, they can't hear your direct transmission if you are on a duplex channel, because you are transmitting on frequency "A" while they are receiving on frequency "B".
This is where repeaters come in. A repeater is an automatically controlled transmitter and receiver that simply transmits what the receiver hears simultaneously. A duplex repeater receives a transmission on one frequency and simultaneously retransmits it on another. The result is that the re-transmission is heard at the same time that the original transmission was made.  
People using a repeater get much greater range from their radio equipment than would be possible talking from radio to radio. Repeater configurations are common in the US. It means that larger area can be covered with duplex system than simplex system.