To burn a card is to remove the top card from the still-undealt portion of the deck and place it, face down, on the table out of play. When used as a noun, the burn is a card that is burned, or is the pile where such cards are placed on the table. The term comes from the idea that you are removing the card from the deck, as though you were literally "burning it up" so it no longer existed. A card being burned is called a burn card.
In nearly all poker games, the dealer burns a card from the deck just before dealing the card(s) that start a new betting round. The exception is the very first deal, prior to the betting rounds, when the dealer is dealing the first cards off the deck, at which time the dealer does not generally burn a card prior to the deal. In other dealing rounds, the dealer burns the top card from the deck, and then deals the card or cards needed for the round.
The historical reason behind having a burn card is much less applicable today, though it still exists. Burn cards help thwart a particular form of cheating using marked cards. When a dealer has completed a deal, the players have a long period of time during which they may see the back of the card which is on top of the deck and would be the first card to be dealt in the next round. If that back were marked in some way, a player who knew about the marking could know which card it was, and would therefore know a card that would get dealt into someone's hand (or into the community cards, in some variants). If the top card is burned instead, then even if the card is marked, the only information a player who knows the markings has is that the card in question will not play, which is a far less valuable piece of information, no worse than a player revealing their cards as they fold (which is against the rules but is merely frowned upon, and does not foul the deal).
Burn cards are kept separately from the muck. This is for two reasons: first, in some variants of poker, the burn cards may actually be put back into play in some cases (such as in 7-Stud), in which case it is important that only the burn cards, which have never been seen by any player, are used. Second, the number of burn cards can help the dealer and players remember (or determine) which betting round is in progress, and whether a burn card has already been burnt for this round. Burn cards are traditionally placed under the chips of the main pot in the center of the table, though this is not required.
In handling misdeals, and in dealing in general, one overriding principle involved is to try to affect the players' cards "as they would have been" as little as possible. Thus, it is important that precisely one burn card be burnt prior to each betting round - and therefore, it is important to keep the burn cards separate so that players can confirm that the proper number of burn cards have been dealt at each point in the hand.