The name of the open-source web server software that powers many websites is Apache HTTP Server. Often referred to simply as Apache, it is a robust and flexible server solution widely used for hosting web content. It supports various programming languages and can be customized with numerous modules to enhance functionality. Apache's popularity stems from its reliability, security features, and active community support, making it a staple in web hosting environments. Other notable open-source web servers include Nginx and Lighttpd.