What do the white and red crosses mean?

There are two kinds of crosses on the board, and they mean different things.

White cross

A mark you add yourself to show that a dog can't go in that cell. The game doesn't check whether it's correct, so it's just a note to help you think.

Red cross

Appears automatically when you double-tap a cell that isn't part of the solution. It marks a wrong move, costs you one life, and stays on the board until the end of the level.