| Rebooting the router and unplugging the router have two different effects. Your router is apparently not saving the DHCP assigned addresses when you unplug it. Since a reboot does not disconnect the power, the memory is not lost and it is still able to retain the data. This can get messy if you have more than just a couple of devices on your network. It is possible that there could be an IP address conflict if the router looses the address table and hands out an address currently in use by another computer.
When a PC (or printer, etc) is first turned and networking is established, it advertises itself to the network for any available DHCP server. The server responds with an assigned address from the available pool of addresses. The period in which the PC keeps the address is called the Lease period. About 60% through the Lease, your computer will start to advertise itself again and request that it continue to use the assigned address. If the PC is not able to request an extension on the Lease (powered off, etc.), the address is placed back into the pool of available addresses for future use by the DHCP server.
By using the IPCONFIG command at the command prompt, you can force some of these actions to take place.
IPCONFIG /RELEASE causes the network adapter to give up the address lease.
IPCONFIG /RENEW causes the network adapter to request a new address lease.
IPCONFIG /ALL shows the currently assigned network information.
You may configure your router's DHCP server to provide longer or shorter lease periods. Most common is between 1 and 7 days. If you do not wish to automatically obtain an IP address and wish to assign a static IP address, do not use an address that is within the range of addresses in the DHCP Scope (available address pool). | 
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