First off, I want to point out that many people don’t actually turn their PCs off; they put them to sleep. Unless you changed the settings, many Windows computers only turn off if you explicitly tell them to Shut Down.
Now, my old rig did that when I put it to sleep; the cat would jump up on the desk and turn it on by pressing a key or moving the mouse. However, if you went through a full boot then this is unlikely.
Sometimes a power flicker can do that too. Many PCs have a setting in the BIOS as to how to handle a power outage; on, off, or “last state”. I always have mine set to “off” but you have to manually set that since the default is usually “last state” which means that a sleeping PC will turn on or sometimes “on”, which will cause “any” computer to start up after an outage.
If the “Wake on LAN” is turned on then a signal from your router (like a “keep alive” ping) can turn it back on. Again, many systems have that set to “On” by default and you have to go into the BIOS to turn it off.
@Captain_Fantasy That is the difference between “Power save” and “Power Off”. That little power draw is the price to pay for getting the computer up and running in 10 seconds as opposed to 30–60 when you turn it on next.
@slick44 I think that there was more to your problems than that. No virus I can think of will prevent a computer from doing a POST; they generally don’t do anything until the computer has at least read something from the hard drive, like trying to boot an operating system.