To mete something out means to distribute or give it in measured amounts, often in a careful or strict way. When a judge metes out punishment to criminals, she decides what penalty each person deserves based on their crime. When a teacher metes out consequences for misbehavior, he's carefully deciding what's fair for each situation.

You almost always see mete paired with out: someone metes out justice, rewards, or punishment. The phrase suggests thoughtfulness and fairness, like a parent who metes out weekly allowance based on completed chores, or a coach who metes out playing time according to each player's effort and skill.

Notice that we usually mete out serious things: justice, punishment, consequences, or rewards. You wouldn't say someone meted out candy at a party (you'd just say they handed it out or distributed it). The word carries weight and suggests that what's being given matters, and that the person giving it is being deliberate and fair.