a phase transformation

Water freezing into ice, or steam condensing into water, is the same stuff rearranging into a different form. Inside a solid metal, a phase transformation is when the atoms rearrange from one crystal arrangement or mixture of phases into another, usually because you changed the temperature. The dramatic part: unlike water-to-ice, which happens right at 0 degrees C, a solid transformation can be sped up or almost frozen out entirely depending on how fast you cool.

A phase is a region with a uniform structure and composition, like ferrite, or austenite in steel. A phase transformation converts one set of phases into another, for example austenite (FCC iron holding lots of dissolved carbon) decomposing into ferrite plus cementite. It happens in two steps: nucleation, where tiny stable seeds of the new phase appear, then growth, where those seeds enlarge. Both steps need atoms to move, and moving atoms takes time, so the transformation is not instant.

The equilibrium phase diagram tells you what phases you should get if you wait forever, but it says nothing about time. Because real cooling is fast, the structures a metallurgist actually uses (fine pearlite, bainite, martensite) are non-equilibrium and never appear on the phase diagram. Controlling the transformation through temperature and time is the single biggest lever an engineer has over a metal's strength and toughness.