The Agile Manifesto is not a method, but rather a philosophy that attempts to do away with the rigid, locked-in methodologies they experienced in software development. Which also lies in the word 'agile', which in Danish can be translated as 'adræt'.

The manifesto reads as follows:

We uncover better ways to develop software
by doing it and by helping others to do it.
Through this work we have learned to value:

Individuals and collaboration rather than processes and tools
Well-functioning software rather than extensive documentation
Cooperation with the customer rather than contract negotiation
Managing change rather than sticking to a plan

There is value in the points to the right,
but we value the points on the left, higher.

The manifesto is also based on 12 principles, which include is about putting the customer at the center, accommodating change, enabling early feedback and continuous delivery, and about the importance of face-to-face communication.

As you can see, both the manifesto and the principles stand in stark contrast to the Waterfall Model, where tasks are solved one by one, like a waterfall:

As I said, the manifesto and the 12 principles are not a method, but rather a philosophy. So how you live up to these is entirely up to you. But anyone who follows the philosophy is, so to speak, agile.

The agile methods have spread
The agile manifesto does not in itself provide a procedure for how to do it in practice. There is, however, a wide range of methods that give an idea of how to put the agile manifesto's philosophy into practice. Here you can e.g. Mention Scrum, Lean, Kaban, Extreme Programming and others.

The agile form of project management has proven to be an effective work process, and agile companies, projects and teams are now found within a wide range of industries.