The production designer is the person responsible for creating the overall visual look of the film, television show or theatre production. If you're watching a film set in autumn, and you notice recurring colors of dark green, navy and brown, it's because that's the look the designer intended. If you're watching a production about a reclusive woman and the set of her apartment is covered with hundreds of books, trash, an unkempt house and worn furniture, it's because that's what the designer intended. The entire visual look of every scene falls under the purview of the production designer.
The art director works directly under the production designer, but often the relationship is more collaborative than hierarchical. The art director assists the production designer in creating the overall visual look of the project, but in a more managerial and budgetary nature. The art director is responsible for staffing the department with dressers, decorators, painters, carpenters and assistants, in addition to overseeing the budget for the department. The art director files expense reports with the production office, prevents budget overruns and assists the production designer find within-budget set pieces.
The production designer leaves it to seasoned, professional painters and carpenters to physically create the intended visual set pieces. Many times, carpenters will assist in finding a solution, both financial and creative, to producing the designer's intended visual. For example, some set pieces are too expensive to produce using standard materials, like wood, and carpenters may suggest using sculpted Styrofoam to produce the same visual for less money.
Unlike the art director, the production designer, painters, carpenters, set dressers and decorators join projects later in production. They assist during principal photography to ensure sets are established, recreated if necessary, and decorated appropriately. Their ability to intuitively know what should and should not be placed in scenes is invaluable, but much of their direction comes directly from the production designer.