A gag is something such as a piece of cloth that is tied around or put inside someone's mouth in order to stop them from speaking.

If someone gags you, they tie a piece of cloth around your mouth in order to stop you from speaking or shouting.

If a person is gagged by someone in authority, they are prevented from expressing their opinion or from publishing certain information.

If you gag, you cannot swallow and nearly vomit.

A gag is a joke.

If you say that someone is gagging for something or is gagging to do something, you are emphasizing that they want to have it or do it very much.