A stile is each of the upright pieces of the frame or leaf of a door or a window. Together with the rails, which are the horizontal pieces, it forms the structure that holds them together.
On a door, the hinges go on one stile and the lock on the other, with its strike plate on the frame. On an aluminium or PVC window, the catch or the handle goes on the stile.
That is why it has so much bearing on security. A weak stile or a frame badly anchored to the wall lets the door give way to a crowbar, even if the lock is a good one. If your door moves in its frame or has gaps, take a look at door reinforcement.