T568 is the technical name for the pin out of an RJ-45 cable. There are two types, T568A and T568B. A crossover cable has one of each end, while a straight through cable is the same on both ends. A straight through can be both T568B or T568A.
An Ethernet crossover cable is a crossover cable for Ethernet used to connect computing devices together directly. It is most often used to connect two devices of the same type, e.g. two computers (via their network interface controllers) or two switches to each other. By contrast, straight through patch cables are used to connect devices of different types, such as a computer to a network switch.
Intentionally crossed wiring in the crossover cable connects the transmit signals at one end to the receive signals at the other end.
Many network devices today support auto MDI-X (aka "auto crossover") capability, wherein a patch cable can be used in place of a crossover cable, or vice versa, and the receive and transmit signals are reconfigured automatically within the device to yield a working connection.