A general term used to describe a more flexible approach to work which moves away from stage based projects as these are often slow and resource heavy (see Waterfall). Agile aims to provide a more flexible type of change that is document and person light. To achieve this, projects are typically managed through a series of smaller tasks.
The term originated from software development. More specifically, from the Manifesto for Agile Software Development which was developed as a set of principles which would underpin good projects in 2001. It is now used as a broad name for a range of software development methodologies such as scrum and Kanban and Extreme Programming (XP).
It is incredibly popular to call your project or even your whole business ‘agile’. Software development firms also often refer to themselves as being agile development shops. Unfortunately, many companies focus on the speed of delivery and lack of documentation, without understanding the rigour, peer review and thought through architecture which is needed to make a success of agile approaches.
This is more like fragile development, and it often results in poor quality software being delivered for high cost in programmes which carry a high project risk and little to no governance.