You've gotten a lot of comments on this, mostly because you've asked a very open-ended question. Let me try to offer my interpretation to the question.
There are a bunch of active parts on the mainboard even when the power is off -- for example, your clock. You wouldn't want to reset the clock every time you power off, or even every time you unplug -- it lives on a battery while you're not looking. Some things like BIOS passwords also lives on that same battery -- so, at least, it used to be the case that when you wanted to bypass someone's boot password, you would break into the case, pop out the battery, unplug, wait five minutes to make sure that everything is dead, pop the battery back in, and start up from scratch. It's been a while since I last had to do that though. ^_^
The moment you press the power button, what you are really doing is completing a circuit between two pins exposed on the mainboard. Your power button literally has wires connecting to a socket which sits atop those pins. One of those pins has a voltage, one of them expects it, so the very first thing that happens is, "a burst of current flows from pin A to pin B." The motherboard is an electronic system which is already "on", and which is expecting this and waiting for it. (And you can trigger it with a screwdriver in a pinch.)
So that is "the thing that tells the firmware to initialize." To be fair, it might have a whole cascade of effects -- it might have to trigger the power supply or so, has to get the fan on the CPU running and then the CPU itself needs to start from a predictable state, and the mainboard needs to load its own firmware -- but the basic idea here is to just have a circuit which knows what this one little current pulse means.
There are a bunch of active parts on the mainboard even when the power is off -- for example, your clock. You wouldn't want to reset the clock every time you power off, or even every time you unplug -- it lives on a battery while you're not looking. Some things like BIOS passwords also lives on that same battery -- so, at least, it used to be the case that when you wanted to bypass someone's boot password, you would break into the case, pop out the battery, unplug, wait five minutes to make sure that everything is dead, pop the battery back in, and start up from scratch. It's been a while since I last had to do that though. ^_^
The moment you press the power button, what you are really doing is completing a circuit between two pins exposed on the mainboard. Your power button literally has wires connecting to a socket which sits atop those pins. One of those pins has a voltage, one of them expects it, so the very first thing that happens is, "a burst of current flows from pin A to pin B." The motherboard is an electronic system which is already "on", and which is expecting this and waiting for it. (And you can trigger it with a screwdriver in a pinch.)
So that is "the thing that tells the firmware to initialize." To be fair, it might have a whole cascade of effects -- it might have to trigger the power supply or so, has to get the fan on the CPU running and then the CPU itself needs to start from a predictable state, and the mainboard needs to load its own firmware -- but the basic idea here is to just have a circuit which knows what this one little current pulse means.