How pieces move

Basics

White moves first. The goal is to put the opposing team's king in checkmate. The game is not lost or won until this. Even if you only have your king left, you can still play.

Putting the king in checkmate is done by alternating turns, each turn you can move one piece.

Terms

Capture: "Attacking" an opposing piece with your own. This is done by moving your piece to the same square as the opposing with a legal move. Your piece will replace the other, which will be removed from the board.

Square: One spot (or square) on a chessboard.  You can classify a square by looking at the side of the board

Checkmate: When a King cannot move to a square where it is safe from being captured.
 
Fianchetto: A move when the pawn on B2 or G2 (white) or when the pawn on B7 or G7 (black) is moved one space forward and is filled by a Bishop the next turn.

Castling
:
A special move which can only be done if you haven't moved your king and either Rook. The king moves two spaces towards the rook, and the rook moves in the space between the current and former location of the king.