Four values, not four rules
The Agile Manifesto, published in 2001, comes down to four preferences: 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. The text states that the items on the right still have value. It does not say "no plan" or "no documentation": it says what comes first when you have to choose.
Ten years later, a special issue of the Journal of Systems and Software took stock of the research and was still trying to explain why these practices work, rather than just describing them (Dingsøyr et al., 2012). In other words, agility is a set of practices that proved themselves in the field, with the theory arriving afterwards.
What it changes day to day
Work becomes visible
You cannot adjust what you cannot see. The first move of a team becoming agile is often to put all its tasks on a shared board, even if that reveals ten things "in progress" for three people.
The cycle gets short
Instead of planning twelve months, you give yourself one or two weeks, one objective and a demo at the end. That is the basic principle of the Scrum method, though you can apply it without adopting the whole framework.
Measurement stays simple
At the end of a cycle, two or three numbers are enough: what is done, what is blocked, what slipped. The point is to discuss them, not to grade the team.
Starting this week with a team of five
- Monday: everyone lists their current and upcoming tasks, unfiltered. They go on a shared board.
- Pick a cycle length (one week to begin with) and a single, realistic objective.
- Every day, five to fifteen minutes: what is moving, what is blocked.
- Friday: thirty minutes to show what is truly done, then say out loud what changes next week.
If the objective is missed after two cycles, first look at task size and estimation before looking for another method.

Sources
- Beck, K. et al. (2001). Manifesto for Agile Software Development. agilemanifesto.org
- Dingsøyr, T., Nerur, S., Balijepally, V., & Moe, N. B. (2012). A decade of agile methodologies: Towards explaining agile software development. Journal of Systems and Software, 85(6), 1213-1221. doi:10.1016/j.jss.2012.02.033