the hyperbolic metric
A metric is just a rule for measuring lengths and distances. The hyperbolic metric is a different rule from the ordinary Euclidean one — a rule under which the unit disk becomes an entire non-Euclidean world, the hyperbolic plane. Under this metric, the closer you get to the rim of the disk, the more 'expensive' each step becomes: a path that looks short near the boundary is actually enormously long in hyperbolic terms, and the boundary circle itself sits infinitely far away. Living inside the disk with this ruler feels like living in an infinite plane that just happens to be drawn shrinking toward a finite edge.
Concretely, the hyperbolic metric assigns to a tiny step of Euclidean length |dz| at the point z a hyperbolic length of 2 |dz| / (1 - |z|^2). The density factor 1/(1 - |z|^2) is 1 at the center and grows without bound as |z| -> 1, which is what stretches distances near the rim. To find the hyperbolic length of a curve, you integrate this density along it; the hyperbolic distance between two points is then the smallest length over all connecting curves. This metric has constant negative curvature (curvature -1 with the factor of 2; conventions vary), the geometric signature of hyperbolic space. Crucially, the conformal automorphisms of the disk are EXACTLY its isometries — they preserve hyperbolic length perfectly — which is why the disk is a homogeneous (everywhere-the-same) model of hyperbolic geometry.
The hyperbolic metric is the geometric meaning hidden inside the Schwarz-Pick lemma: that lemma simply says holomorphic self-maps of the disk never increase hyperbolic distance. It is the natural stage for hyperbolic geometry, the modular group, Riemann surfaces of negative curvature, and much of complex dynamics. A caveat worth stating: the disk and the upper half-plane carry the SAME hyperbolic geometry — they are isometric via the Cayley transform — only the formula for the density looks different (on the half-plane it is |dz|/y, using the imaginary part y). And 'hyperbolic distance' is genuinely a distance (it satisfies the triangle inequality), but it is not the Euclidean distance you see with your eyes; the picture deceives, the metric does not.
Near the center the two rulers nearly agree: at z = 0 the density is 1/(1 - 0) = 1, so a tiny step there costs about its Euclidean length (times the factor 2). But at z = 0.99 the density is 1/(1 - 0.9801) is about 50 — each Euclidean inch counts as roughly 100 hyperbolic inches (with the factor 2). Walk toward the rim and the cost runs to infinity, which is why you can never actually reach the boundary in finite hyperbolic distance.
Density 1/(1 - |z|^2): tame at the center, exploding at the rim — the boundary is infinitely far.
Conventions differ by a constant factor (some write 2|dz|/(1 - |z|^2), some |dz|/(1 - |z|^2)), which rescales all distances and shifts the curvature between -1 and -4. The geometry is identical; only the unit of measurement changes. Always check which normalization a source uses.