In magic, a convincer is any action intended to convince the spectator of something that furthers the method. If you've ever spread a deck of cards face up to show they're all different before letting a spectator choose one, congratulations; you've used a convincer in your magic.
Convincers do not always need to be lies. You might be showing the spectator that they genuinely had a free choice of cards. Sometimes, a convincer will involve its own kind of method in itself. For example, you might want to rattle a cup with a coin inside before vanishing it, and the fake sound of the coin rattling inside acts as a convincer, but that's a trick in and of itself.
You might wave your hand over a floating dollar bill to prove that there are no strings above or below it, and this acts as a great convincer. But to pass a solid ring over the floating bill as a convincer, you'll likely need to rely on an additional method like a gimmicked ring or sleight-of-hand to pass the ring over it.
Successful magicians use different types of convincers, and the more you learn about each type, the easier they are to apply to your magic. Almost every trick can be enhanced with convincers. Some convincers are more enhancing than others – for example, telling a spectator that a card box is empty isn't as convincing as shaking out all of its contents in front of the spectator.
There's a type of convincer in magic called a low-odds convincer, and it's absolutely genius and so satisfying. This form of convincer is left to chance in some ways, which is why it's so fooling. Even so, there's also an element of multiple-outs (another great magic principle), which means the low-odds convincer works every time.