phase equilibrium
Drop an ice cube into a glass of water at exactly the freezing point and, remarkably, nothing happens: the ice neither grows nor melts. Two distinct phases — solid and liquid — sit together indefinitely, each perfectly content. Phase equilibrium is this balanced coexistence, and thermodynamics tells you exactly the conditions under which it can occur.
For two phases of a substance to coexist stably they must be in three kinds of balance at once: thermal (equal temperature T), mechanical (equal pressure p), and chemical or diffusive (equal chemical potential mu for each species). The last condition, mu_1(T,p) = mu_2(T,p), is one equation in the two variables T and p, so it defines a curve in the p-T plane — the coexistence (or phase-boundary) curve. Physically, equal chemical potentials mean there is no net drive to convert one phase into the other.
Phase equilibrium is the foundation beneath phase diagrams, the Clausius-Clapeyron equation (which is just the slope of that coexistence curve) and the Gibbs phase rule (which counts how many phases can meet). At a triple point all three conditions coincide for three phases at once, fixing a single point. Caveat: coexistence is a statement about stable equilibrium; metastable states like supercooled water can persist far from it until a fluctuation tips them over.
At the triple point of water — 273.16 K and 611.7 Pa — solid, liquid and vapour all have the same chemical potential and coexist simultaneously. Change T or p even slightly and at least one phase becomes favoured and the others begin to disappear; the triple point is therefore a single fixed point, not a line or area.
At the triple point three phases share one chemical potential and coexist at exactly one T and p.
Equality of chemical potentials, not equality of energies, is the true condition for coexistence; a phase can be the stable one even at higher internal energy if its entropy makes its Gibbs energy (chemical potential) lower.