Definition of burn
The expression is often used to refer to "burning a card", i.e. removing it, discarding it. The shuffling technique will vary depending on where you are playing, but it often works like this: The dealer shuffles all 52 cards. When he has finished shuffling, he presents the deck to the player on his right. He presents the deck to the player to his right because he is the last player to receive his cards. This player cuts the pack, i.e. he will separate it into two parts, equal or not. The second part of the pack will be put back on top of the pack to change the order. When this is done, the first card on top of the deck is burned, i.e. it is thrown into the discard pile. The player to the left of the dealer will then receive the second card.
Burning a card, an additional security
All of these techniques are safeguards to ensure that no one cheats or is able to rig the deck to benefit themselves or a friend. Burning a card makes it harder to cheat. Even in some casinos that have automatic shufflers, the first card is often burned. Some dealers will also burn a card before dealing the turn and river.