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Burning is the process of destroying some of your tokens.
In the old days, this was done by sending them to an unexistend (or at least unclaimed, as far as known) address. Nowadays, some cryptocurrencies provide special 'burn' calls that can be used instead (and are easier for other parties to prove by that the tokens were burnt).
But wy do people burn their Bitcoin or other cryptocurrencies?
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Counterparty coin is one instance I can recall that they burned 20+ bitcoins to prove they are "serious" about their project and from those burned coins they generated counterparty tokens. These tokens can be seen as proof that they are linked to burned bitcoins, but obviously doesn't give them the same bitcoin value. It's more of a: "Hey, I'm serious about my project. Here's proof that I'm going to work on my project by throwing away money, but I'll also generate new coins in its replacement".
