authorize
To give official permission or power to do something.
To authorize means to give official permission for something to happen. When a teacher authorizes a field trip, she approves it and makes it official. When parents authorize a doctor to treat their child, they're giving formal consent. The principal might authorize the purchase of new computers for the library, meaning she has the power to say yes and make it happen.
Authorization usually comes from someone with authority: a person in charge who has the right to make decisions. You can't authorize something unless you have that power. A student can't authorize a school assembly, but the principal can. A bank manager can authorize a large withdrawal, but a teller might need permission first.
The word carries a sense of official approval with formal weight and consequences. If your friend says you can borrow their bike, that's casual permission between equals. But when a government agency authorizes the construction of a new bridge, that's a formal legal approval with documents and signatures. Congress must authorize how the government spends money. Scientists need authorization before conducting certain experiments.
When something is unauthorized, it means no one with authority approved it. An unauthorized party means someone threw it without permission from whoever was in charge.