The significance of the Agile Manifesto in project management is its promotion of an iterative approach to software development that emphasizes collaboration, flexibility, and customer feedback. Established in 2001, the manifesto values individuals and interactions over processes and tools, working software over comprehensive documentation, customer collaboration over contract negotiation, and responding to change over following a plan. These principles drive efficiency and adaptability in project teams, fostering innovation and facilitating quicker adjustments to meet evolving project requirements.