John Lilly became chief executive of Mozilla in January, moving up from his role as chief operating officer. He’s been with the company that created the open source Firefox browser since 2005, the year Firefox 1.5 was released. Before Firefox, Microsoft’s Internet Explorer dominated the Web. Now Microsoft’s share is down and Mozilla’s share is 20 percent. The size of the organization makes Mozilla’s tremendous success that much more remarkable. Headquartered in Mountain View, Calif., it has fewer than 200 employees.