When your opponent makes a double-start move from the home row of pawns (from the starting row for pawns, a pawn may advance 2 spaces as its first move) such that it ends up alongside a waiting pawn of yours, on your next turn, you may capure the opposing pawn that sneaked past yours by selecting your pawn as the piece to move, then selecting the space just past the opposing pawn (it will be diagonally behind the opposing pawn) as the destination. Your pawn will be moved to that position, and the opposing pawn will be eliminated, just like in real chess!
When you advance your pawn all the way to the opposite edge of the board, you will be allowed to trade your pawn for a queen, rook, bishop, or knight.
To castle, select your king as the piece you want to move. If the conditions are such that a castle is possible, you will be able to click on the space 2 spaces away from the king, either on the left or the right side. Doing so will automatically castle the king, moving the king and the rook on that side into the proper position.