Yep, you’ve got it right. The power factor is the ratio of the “real” power to the total apparent power. What it works out to is that the real power is the voltage multiplied by the current multiplied by the cosine of the angle between them. The “power factor” is then the cosine of the phase angle.
To expand on this a bit (for the benefit of those who don’t know much about the subject, since this is a basic level thread), if you take something like a coil of wire (a simple inductor) and run electricity through it, the coil stores energy in a magnetic field. When you remove the electricity, the magnetic field collapses, releasing the stored energy as electricity again. So if you apply a sine wave voltage to a coil of wire, the coil is going to charge up during the first part of the sine wave, then discharge its energy later in the sine wave. If you take two plates of metal that are close together (a simple capacitor) then the same kind of thing sorta happens. The plates store energy, but they do it in an electric field instead of a magnetic field.
In power systems, where the voltage is a sine wave, it ends up that inductors and capacitors work opposite of each other. While inductors are charging capacitors are discharging, and when the inductors are discharging the capacitors are charging. Most homes tend to be slightly inductive overall, due mostly to things like motors in appliances and hair dryers and such.
The power company doesn’t want to have inductive loads, because that means their generators have to waste energy charging up those inductors, only to have the inductors release the energy later. It’s a bunch of wasted energy. Because inductors and capacitors work kinda opposite each other, though, what the power company can do is add banks of capacitors at the substation that they can switch on and off of the line as needed to balance everything out. When they get it balanced just right, the inductors and capacitors (aka “reactors”) balance each other out, so the capacitors store energy and release it to the inductors when the inductors need to charge up, and then when the inductors release their energy it gets stored again in the capacitors. The reactive energy basically just bounces back and forth between the inductors and capacitors, and the power plant’s generators only need to produce the “real” power.
If they use capacitors to balance things out, then there isn’t any additional load on the generators.
However, those capacitors aren’t free. For residential loads, the power company usually just considers them to be the cost of business and they don’t bother to even monitor your power factor. For commercial and industrial customers though, those capacitors can end up being very large and very expensive. The power company therefore monitors the power factor, and if the load is too inductive (which commonly happens with large motors) then the power company charges them out the wazoo for it. This gives commercial and industrial customers a very large financial incentive to do their own power factor correction.
Residential power factor correction devices are advertised a lot with the claim that they will save you money by eliminating the vars (var = volt-amp reactive). Since in almost every residential power system you only charged for the watts (the real power) and not the vars (the reactive power) these devices are claiming to save you money by eliminating something that you aren’t charged for. Needless to say, since you aren’t charged for the vars in the first place, these devices won’t save you any money. If you look inside of these devices they are usually just capacitors, and aren’t switched by any sort of control circuit. Since they don’t monitor the power factor, they can’t actually provide accurate compensation for it. So not only do they not save you money, they don’t even properly do what they claim to do. Avoid devices like this. They are basically scams.