Wi-Fi Direct refers to a way of establishing a direct Wi-Fi connection between two Wi-Fi enabled devices, rather than between a Wi-Fi enabled device and a Wi-Fi access point. That means, it can be used to set up a Wi-Fi network between two Android devices (or other devices supporting Wi-Fi Direct) even when no Wi-Fi access point is available.
Not all Android devices support Wi-Fi direct. Some manufacturers may support it under another name, or may support an alternative and competing technology instead. Others may simply lack support.
Support for Wi-Fi Direct within Android is still relatively poor and despite the presence of some compatible Android apps that can take advantage of this feature, it can still be a bit difficult to set up. To fully take advantage of this feature usually requires specific apps.
The basic idea is that once you enter the Wi-Fi Direct Menu, your device starts broadcasting that it supports Wi-Fi Direct to nearby devices, and at the same time starts looking for nearby devices that support Wi-Fi Direct. If a nearby device supporting it is found, you can then establish a connection from one of the devices to the other, through the sharing of a PIN code. Once the connection is established it becomes a two-way connection acting much like as if the two devices were on the same local network.