The device definitions come from the 802.14.5 MAC used in ZigBee. You can disregard the nonbeacon part, as in ZigBee the MAC is always used in nonbeacon mode.
Coordinate: This is the device that starts the network, in MAC layer it performs a passive scan (just turns on the Receiver and looks for energy) to find a quite channel. It then performs an active scan to find other 802.15.4 networks (by sending a beacon request) so it can choose a unique PAN ID. At the Network Layer it also chooses a random network key. From then on the network is started and power cycling the Coordinator will cause it to use the same MAC and Network settings. The Coordinator is also the trust center. So when new devices join the network it distributes the network key (in HA/SE this is encrypted with the Trust Center Link Key defined in the spec). if the Coord is turned off devices can not join the network, but other routing between existing devices ca continue. Coordinator is typically a mains powered device and does not sleep. There is only 1 Coordinator per network. Other than this it behaves the same as a router.
Router: The router can allow devices to join the network through them, but the trust center messages are "tunneled" to the joining devices from the Coordinator. Routers as the name suggests route messages through the network. Routers are typically a mains powered device and does not sleep. Routers queue messages for their EndDevice "Children".
EndDevice: The EndDevices have no routing responsibilities, they have a "parent" router that queues messages allowing them to sleep. EndDevices are typically battery powered devices.