Ask ten agilists what Agile is all about, and likely you'll get ten different answers.
I would say Agile is all about feedback. Other Agile processes, practices and principles follow from the Inspect & Adapt cycle.
Arrange, Act, Assert
If you are familiar with writing unit tests you probably know the Arrange, Act and Assert pattern. Sometimes it's called Setup, Execute and Validate. It amounts to the same thing, but AAA is a nicer acronym, isn't it?
A unit test is a small piece of code that tests a single assumption. The first thing a unit test does is setting up the data and objects in such a way that it can perform the test. This is the Arrange part. Then it executes the actual function you want to test. This is the Act part of the pattern. Finally, the it checks whether the result is what you expected, which is the Assert part of the AAA pattern.
Unit tests validate the functionality of the software at a microlevel. There are many more such patterns to be found.
For example, a user story usually comes with a set of acceptance criteria. When the story is reviewed by the customer she will check whether the story meets these criteria. In a sense, the user story and the accompanying acceptance criteria are the Arrange part, the actual building of the functionality is the Act part, and the customer review is the Assert part.
Scrum facilitates this kind of review by prescribing the Sprint Review meeting, where the goal is to gather feedback on the product increment from stakeholders, such as customers and users.
Another Scrum meeting that facilitates the Inspect & Adapt cycle is the Sprint Retrospective. Here the process is inspected rather than the product, but the principle is the same. At the Sprint Retrospective plans for improvement are made that are carried out in the next sprint. At the next Retrospective it is checked how these improvements have worked out. This may lead to new improvement plans, and this cycle is repeated over and over.
The Plan-Do-Check-Act cycle was made popular by Dr. W. Edwards Deming in the 1950s. Originally used in manufacturing, it is another example of the Inspect & Adapt cycle. During the Plan phase the output expectations are established. Then this plan is executed and the product is made during the Do phase. It is checked whether the results match the expectations. Appropriate actions are taken in the Act phase, and this cycle is also repeated.
A variant of the PDCA cycle is called Hypothesis, Experiment, Evaluation, which is used in the Lean Startup method, made popular by Eric Ries. First, a business hypothesis is formulated. Then, as quickly and as cheaply as possible, this hypothesis is implemented and evaluated. If the outcomes are not as expected a business "pivot" may be needed, steering the company in another direction, and the underlying assumptions may be adjusted. Of course, this is then again subject to the same cycle, over and over again.
As you can see, there are many feedback loops at many different levels. Combine some or all of them to continuously improve your processes, products, services and businesses.
I'm sure there are many more examples of feedback loops. Feel free to leave a comment to let me know which ones I missed.