From help.ubuntu.com/community/WakeOnLan

In order to use WoL it must be supported and enabled as necessary in the BIOS, NIC and other network hardware (routers, switches etc). You may also need to configure some software if things don't work out-of-the-box.

Does this mean that I must have WoL enabled in all 3 items or just one of them?

My question is because the NIC says:

Supports Wake-on: pumbg

Wake-on: g

Which, means that WoL is present and enabled. However, there is no option in the BIOS (it's a eeeBox 1503) to enable WoL. Actually, it loads a simple UEFI, not exactly the BIOS, so that is probably why I don't see any advanced option.

The router supports WakeOnLan. So, the BIOS is the only item I can't enable it. Will I need to find a way to enable it there, or having it enabled on the NIC is enough?

EDIT

It was not working because I was trying to power up when the machine is completely off and not wake up from sleep/suspend mode. I want to power up, but I'm not sure if this is possible.

The answer provided by psusi answers my question. Using ethtool to enable WoL is enough.