Basically, when your machine is plugged in, even turned off, some maintain some very basic functions such as the network card.
They can be configured so that when a special instruction is received, it can power up the machine.
The easiest way to start is simply, if you have an integrated NIC, look in the BIOS and see if it supports it. If you have an actual Network card, you may not be able to use it... But it is worth seeing if you have a slot for a Wake On Lan Chip or cable
Also, as for saving energy, you need to obviously turn on your device! You need to have a router that can turn on devices based on a schedule or a wireless mobile or similar that you can send the instruction from - obviously though, you may just be better off turning the machine on via its power button!
Wikipedia has a good article on Wake On Lan
Wake on LAN (WoL) support is implemented on the motherboard of a computer and the network interface, and as such, is not dependent on the operating system running on the hardware, although the operating system can sometimes control the WoL behaviour. If the network interface is a plug-in card rather than being integrated into the motherboard, the card may need to be connected to the motherboard by a cable. Motherboards with an embedded Ethernet controller which supports WoL do not need a cable.
Best Answer
You'd have to have a second, awake computer (or WoL-capable router) on the LAN to send the magic packet. It is Wake-on-LAN, after all.
The router method is easier. Check out this DD-WRT article to get it going.