The pawns

It makes most sense to start with pawns as they are so basic most people don't consider them pieces. Pawns can move one square forward or capture diagonally except on their (the pawn's) first move when they can move two squares forward. Also if your opponents pawn has moved two squares forward so that it is next to your pawn, on your turn you may capture it as if it had only moved one square forward i.e. you may take it off the board and move to the place one square behind the pawn it captured. This called En passant-
 

Pawns are useful for attacking two pieces at once so that your opponent can't save them both. Also they are good for stopping  your opponents king from running away. As well if a pawn reaches the far side of the board it can turn into any piece you want except a king or a pawn. This is called promoting.

Knights

Knights move in L's, two squares in one direction(not diagonaly) and one at a right angle from the two squares. Knights are also the only pieces that can jump over other pieces. Knights are useful for attacking two pieces at onces and attacking pieces behind other pieces.

Bishops

Bishops can move diagonally as many squares as they want. Note that at the begining of the game one bishop is on the white squares and one is on the dark. These bishops can never go on the other color. Bishops are useful for defending and checkmating.

Rooks

Rooks move horizontally and vertically as many squares as they want. They are good for attacking late in the game. They also are good to place behind pawns you're going to promote. Also they are used for castling.

Queens

Kings

Kings are the most important pieces on the board because if your king is trapped and attaked you lose. You can sacrafice anything if it will save your king and it will be a good move. Also it is good to remember that if your opponent is winning you should try to get into a postion where you can't move because then the game is drawn (a tie). Now, then kings can move one square in any direction as long as they won't be attacked on that square. Also kings have the move called castling. This is where you move the king two squares to the right or the left then you move the rook to the oppisite side of the king. However the king and the rook must not have moved, there must be nothing between them, and finally the king can not move through or into check.

Before castling


After castling

The starting position