The game goes through two distinct stages. During the first stage each player joins dots, being careful not to complete the third side of a box for their opponent, until the cells are divided into one or more chains or loops, like this:

When beginners play, the first player forced to complete the third side of a box chooses the shortest chain, and the players then complete the chains in turn.

For example, in this case Red starts the chain of four boxes, Blue takes them all, and then has to start the chain of 5 boxes for Red:

The result is that Red wins by 5 boxes to Blue's 4.

However a simple insight allows Blue to win in this example. Instead of taking all 4 boxes in the first chain, Blue can decline the last two boxes. Red is then forced to take these, and start the chain of 5 boxes for Blue:

The result is that Blue now wins by 7 boxes to Red's 2.

The secret is to decline the last two squares of every long chain of three or more squares. You will then win all but two of every chain, plus the whole of the last chain.

This is just one example showing that Dots and Boxes is a much deeper game than many players realise.