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Promotion;
The pawns have an ability to change; if they reach the other side of the board they can become another chess piece, any piece. There is the common misconception that a pawn can only be exchanged for a piece that has been captured, that's not true. A pawn is usually promoted to a queen, and specifically pawns, as they are the only piece capable.

En Passant;
Which is French for "in passing". It applies to when a pawn moves out from its first position, moving two squares, and if it does this and lands side by side with an opposing pawn, meaning that it can;t be captured. Except that it can, by using the "en passant" rule, in the move immediately following the opponents pawn's two square jump move, you can capture it as it passes by.

Castling;
This can allow you to both get your king to safety, and get your rook out of the corner and actually into the game. Its achieved by one player moving the king two squares to one side, followed by the rook moving from that side's corner to the right next to the king on the opposite side he moved. this will only work if its both the kings and the rooks first move, that the king isn't in check and there must be no other pieces in between them.

Check & checkmate;
The whole point of the game. Checkmate is achieved when a king is put in check and cannot get out. there are three ways a king can escape check; by moving, by blocking the check with another piece, or to capture the threatening piece.