I tried to explain as much as I could using illustrations. If the written directions are confusing, take a look at the pictures. For the more complicated rules, I illustrated them in sequence.
Step 1: Setup, Turns, and Taking Pieces
The board is setup as shown. There should always be a white square at the closest right-hand side for both players. Remember that the queen must be on a square that matches her color.
White always moves first, and players alternate turns. Players can only move one piece at a time, except when castling (explained later).
Players take pieces when they encounter an opponent in their movement path. Only pawns take differently than they move (explained later). Players cannot take or move through their own pieces.
Step 2: Pawn Movement
If a pawn reaches the opposite side of the board, it is promoted to a higher piece (except king). There is no limit to how many pawns can be promoted.
Step 3: Rook
Step 4: Knight
To make it easier to remember how a knight moves think of an L. Two spaces in a direction forward, backward or side-to-side, and one space at a right turn.
Step 5: Bishop
Step 6: Queen
Step 7: King
A king cannot move to a square that is under attack by the opponent.
Step 8: Special Move: Castling
During castling a king moves two spaces towards the rook that it will castle with, and the rook jumps to the other side. The king can castle to either side as long as:
1. The king has not moved.
2. The king is not in check.
3. The king does not move through or into check.
4. There are no pieces between the king and castling-side rook.
5. The castling-side rook has not moved.
It does not matter:
A. If the king was in check, but is no longer.
B. If the rook can be attacked by an opponent's piece before castling.
Step 9: Special Move: En Passant
Step 10: Check
A player cannot move their king into check.
Step 11: Checkmate
A king is in checkmate if it is in check, the opponent's piece that has the king in check cannot be captured, the check cannot be blocked, and the king cannot move to a square that is not under attack.
In the illustration the white queen has the black king in check, and all of the spaces where the king can move can be attacked by the queen. The king cannot take the queen, because the knight is protecting the queen. The black bishop cannot block the queen. This is checkmate.
Step 12: Stalemate
In this illustration it is white's turn. All spaces around the king are being attacked, but the king is not in check, therefore it cannot move. The only other white piece, the pawn, is blocked by the king. Because movement is impossible, the game is a stalemate.
If white had another piece somewhere on the board that was not blocked, it would have to move. The game would continue.
Step 13: Basic Strategy
Obviously you want to protect your pieces from capture, but it helps to know which pieces are the strongest so you can decide who to save if you must choose between two. A good explanation of piece value is available on Wikipedia
Queen: Strongest = Most Value
Pawn: Weakest = Least Value
The bishop and the knight are commonly considered equal on the value scale, however many feel (myself included) that the bishop has a slight edge over the knight.
Pawns become more valuable as they near promotion.
Although a pawn can be promoted to a variety of pieces, the strongest choice is almost always to promote to queen.
When building defenses, remember to look at the board and gauge how strong you are in certain areas of the board. Try an keep power distributed fairly evenly, and bring pieces over to add strength if you see an attack coming.
When attacking, it's a bad idea to let any of your pieces become cut off from your main force. I find it helpful to have a support piece in mind when making an attack. Using pieces in tandem almost always yields a better result than using one piece alone.