To green-light is to give permission or a go ahead to move forward with a project. The term is a reference to the green traffic signal, indicating "go ahead."
In the context of the film and television industries, to green-light something is to formally approve its production finance, and to commit to this financing, thereby allowing the project to move forward from the development phase to pre-production and principal photography.
The power to green-light a project is generally reserved to those in a project or financial management role within an organization. The process of taking a project from pitch to green light formed the basis of a successful reality TV show titled Project Greenlight.