You start out with eight of these so it's worth knowing how they move! Pawns can only move forwards in a straight line. For each pawn's first move they can move two squares forwards (without jumping over anything) from then on they can only move one square at a time. Of course if you only want to move your pawn one square on its first move you can do that instead. Pawns take opponent's pieces by moving forwards diagonally one square to either left or right.
Pawn Promotion
Because pawns can only move forward they would be useless once they got to the other side. This is where pawn promotion comes in. As soon as your pawn reaches the other side of the board you must replace it with a new piece! You can choose any piece apart from a pawn or a king. Almost always people choose to promote to a queen.
Note: contrary to what some people think you can still promote to a queen even if you already have one queen on the board (giving you two queens).
Knights (horses)
Knights can be quite tricky to get to grips with being animals they are unpredictable and can be difficult to keep under control! Knights are the only piece which can jump over other pieces. Knights move in an L shape by moving two squares in any direction (other than diagonally) before stepping one square to the side.
Note: knights can cover a large area around them so it's best to avoid having them on the side of the board where only half of their potential is usable. Here instead of covering eight squares your knight only covers four.
Bishops
Bishops (sit between the knights and the king/queen) can move only diagonally. However they can move any number of squares and can be very mobile. One bishop starts on a black square and one on a white and due to the way they move they will stay on their own colour throughout the game.
It is normal for bishops to be placed somewhere where they cover one of the long diagonals across the middle of the board.
Rooks (castles)
Rooks are one of the simplest pieces to understand yet one of the most powerful. They can move any number of squares in either a horizontal or vertical direction.
Queens
The queen is the most powerful piece you have so make sure you look after it! If you do lose it make sure that you will take your opponent's queen as a result.
The queen combines the moves of the rook and bishop. I.e. moving any number of squares in either a diagonal, horizontal or vertical direction.
Kings
Yes the king is a liability! However in an end game the king becomes a very effective offensive piece. It is after all the only piece apart from the queen which can move in diagonal, horizontal and vertical directions. The difference is that it can only move one square at a time.