How it's Marketed:
Toothpaste ads are quick to inform us that every time you smile you're forcing the world to tolerate your corn-filled shitlog of a grimace and no one anywhere is enjoying it. Indeed, the fact is no matter how much you brush with your tube of Sponge Bob Bubblegum flavored toothpaste, your teeth are still manky, stained pillars jutting forth like the decrepit fingers of the damned. Brooke Shields wouldn't be half the woman she is today if she was walking around with the Crypt Keeper teeth she was born with
Brooke Shields, 1983.
Fortunately, toothpaste manufacturers offer redemption via just a few weeks of vigorous brushing with their cutting edged whitening toothpastes! What a miracle of modern fucking science!
What it Really Is:
Whitening toothpaste, by and large, is regular toothpaste with grit in it. It can be anything from aluminum oxide, which is the main ingredient in chalk used for billiards, to calcium carbonate, the basic ingredient of many antacids. Basically, if it's gritty, it'll scrub shit off your teeth. They could use sand in whitening toothpaste and get the same effect.
Or you could grind your teeth on a big bag of rocks.
Another popular ingredient is hydrogen peroxide; that stuff that mom put on all your scraped knees that fizzed up like nobody's business. Peroxide does have bleaching properties, but in toothpaste is at such a low level its effect is pretty negligible. So your dentists formulated and recommended whitening formula's actual working components could be mimicked at home if you drop your toothbrush on the floor and simply choose not to wipe all the crunchy shit off of it before brushing.