Proof of burn is a protocol in which users send cryptocurrency coins to an address which no one has the ability to recover thereby reducing the supply of coins in circulation. PoB is supposed to address problems involving energy consumption in Proof of Work systems.
Burned coins are coins that were sent to an 'unspendable' address, an eater address, which have no private keys. Anybody can view the amount of coins in the address, but they are not accessible.
The cryptocurrency Counterparty was started with proof of burn in which users destroyed bitcoins to generate a proportional amount of Counterparty's native currency called XCP.
The online peer to peer marketplace OpenBazaar also used proof of burn to enable users to perform Reputation Pledges. These pledges represent a commitment to an identity on the platform as they have expended resources on it. Thus they are incentivized to maintain a positive reputation with that identity or risk wasting the value of the coins burned for that profile.