Half duplex can not send and receive at the same time. It also has algorithms to detect and handle collisions (CSMA-CD). Think of it like a walkie talkie or a CB radio, only on a network.
Full duplex can send and receive at the same time. Think of it as a telephone.
One issue that can arise is sometimes you will see a 100 Mbps half duplex connection which should almost never happen. Since the 100 Mbps connection will most likely not be detecting collisions (it expects to be full duplex), it can be very very slow.
This normally happens when one side is hard set to 100/full and the other side is set to autonegotiate. The side that is set at auto will not be able to sense what the duplex is (because the other side is hard set) and use half duplex.
So, while you will see 10/half quite often on older equipment, be very very suspect if you see 100/half because you are probably looking at a duplex mismatch.