FreeDrama.net, run by playwright D.M. Larson, offers royalty-free plays for schools and community theatres. Includes sections for children's, full-length, short, one-act, and small cast plays. It is requested that theatre groups request formal permission to perform, and send a copy of their program to the playwright.
This website offers links to "E-Scripts" from a variety of time periods and genres. Click "E-Scripts" on the top navigation menu to view a list of options. The list also includes an Electronic Text Archives page which provides resources for further playscript searches. Check carefully for production rights; many links lead to outside websites with their own performance policies.
ProPlay makes plays which have received professional productions available free-of-charge on the Web, to encourage their further production. In many cases, you'll find the complete text of the play here, along with production history, a biography of the playwright and contact information. In other instances, the playwright has chosen to post a portion of the play, and you may contact them or their representative for a complete copy.