When a player hesitates, he expresses doubt about his call, but his partner must not take advantage of this doubt. That information is unauthorised. Even ethical players may take advantage subconsciously, so the Law takes note of this.
Once a player has hesitated, his partner may not choose amongst logical alternatives an action suggested over another by the hesitation. For example, suppose a player doubles after a think: while this is for penalties it is clear that he has doubts. Suppose further that his partner has a hand on which any player might consider whether to leave in or take out: then because of the hesitation the player may not choose to take it out.
When there appears to be a hesitation the Director is needed for two reasons: first to establish whether there was a hesitation, second to decide whether the action chosen was proper. When there is a hesitation the Director should be called. Exactly when? It is normal to call the Director after the partner makes a call that could be affected by the hesitation. However, in North America, it is recommended that the Director is called immediately after the hesitation. The main purpose of the call at that time is to establish whether there actually was a hesitation. Play then continues.
If there might be a problem, the Director is recalled at the end of the hand, and he then decides on the basis given above. If he thinks a player has chosen an improper alternative then he will adjust the score to what it might have been otherwise. His decision may be appealed, and often will be: Appeals Committees are at their most useful in making bridge judgements of this sort.
Players should not worry about being ruled against in such circumstances even if they feel they have acted ethically: no-one has impugned their ethics merely disagreed with their bridge judgement.