Copyright protects authors' basic rights over their work--to perform it publicly, reproduce and distribute it, and make sure its integrity is not compromised. 

A copyrighted play usually is offered as a licensed property. If you wish to produce a play protected by copyright, you apply for a license--in essence, paying the publisher/agent (and thus the playwright) for performances of the play. These payments are referred to as "royalties." 

Producing a copyrighted work requires permission and/or payment of royalties. This includes not only plays and musicals, but background music used in a production or performed on stage as part of a play. This is true whether you charge admission or not, and whether you are a for-profit or not-for-profit theatre or a school.