Agile methodology is an alternative to traditional project management. It is typically used in software development. Agile helps teams respond to unpredictability faster through incremental, iterative work cadences, so called sprints. Agile methodologies are an alternative to traditional sequential development ("Waterfall").
From the technical perspective, Agile is a time-boxed, iterative approach to product delivery that builds the product (i.e. software) incrementally from the start of the project (shipping "usable increments" continuously to the user), instead of trying to deliver it all at once near the end ("Waterfall Delivery"). Scrum is the most popular agile software delivery process framework.
From a wider perspective, Agile is a leadership mindset to organise and manage people in a self-organised way with at least low, better no hierarchies. The team "inspect and adapt" their work regularly: they learn from every change they make, every mistake, every failure.