General Move Rules of Chess
There are some general rules that apply to all the pieces and pawns and moves. Some of these seem obvious, but they still need to be stated.
Captures: You capture an enemy piece by moving your piece or pawn to that square, removing the enemy piece, and leaving your piece on that square. A capturing move of an enemy piece on a square is the same move as a normal move to an empty square for 5 of the 6 pieces: Queen, Rook, Knight, Bishop, and King. The Pawn is unusual in that it has different rules for capturing (diagonally) than for normal moves (forwards/upwards).
Most pieces cannot jump over other pieces or pawns. Only the Knight (Horse) piece can jump. All of the other pieces cannot jump over any pieces, yours or enemy pieces, along ranks (horizonally), files (vertically), or diagonals. Pawns cannot jump pieces when they do their 2-square jump. King castling cannot jump over pieces.
Pieces can move backwards, pawns cannot. Only the Pawns are prevented from moving backwards. All of the pieces (Queen, Rook, Knight, Bishop, and King) can move backwards. When pawns are promoted into a Queen (or other piece), the new Queen can then move backwards. Backward moves are often overlooked by beginners.
Only one piece or Pawn per square: The pieces cannot "share" a square.
Only one square per Piece: Also, a piece or Pawn cannot sit on 2 or more squares, it has to clearly sit in the middle of a single square. If your opponent moves a piece to an unclear square, or its on the line of two squares, you need to clarify what square it is on.
No "pass" move allowed: You cannot just "pass" at making a move. You are required to make a legal move if possible. If there is only 1 legal move, you are required to play it. If there are zero legal moves, then the game ends and it is either checkmate or stalemate. Sometimes you don't have any idea what to play, and sometimes every move is bad, but you still have to choose a move.
Capturing is optional. You are not required to take. Normal chess has no "must take" rule. You can choose to take or not. You can move right up next to an enemy piece if you like, you don't have to take it. There are some odd chess variants that sometimes have a "must capture" requirement (e.g. suicide chess).
There is no "chance" or "surprise" in chess. There is no concept of chance in chess, there are no moves decided by the roll of dice or the flip of a coin. There is also no "surprise" moves in chess. Chess is not like a card game where you don't know your opponent's hand. In theory, you can see every possible move, although sometimes it doesn't feel like it! In a game of chess, you can see all the pieces and know all the moves. It's a pure game of strategy.