a Riemannian metric
Vol I let you measure lengths and angles on a surface sitting in space by borrowing the ambient dot product. A Riemannian metric is the device that lets you do this on a manifold all by itself, with no surrounding space to lean on: it equips each tangent space with its own inner product, smoothly varying from point to point. Once you have it, you can speak of how long a tangent vector is, what angle two of them make, and how to integrate length along a curve.
Precisely, a Riemannian metric g on a smooth manifold M assigns to each point p a positive-definite symmetric bilinear form g_p on the tangent space T_p M, varying smoothly with p. In local coordinates x^1, ..., x^n it is written ds^2 = g_ij dx^i dx^j, where the matrix g_ij(p) = g_p(d/dx^i, d/dx^j) is symmetric (g_ij = g_ji) and positive-definite at every p. The length of a curve gamma from a to b is the integral over [a,b] of sqrt(g(gamma', gamma')), and the angle theta between vectors u, v satisfies cos theta = g(u,v) / (|u| |v|). The pair (M, g) is a Riemannian manifold.
Two honesty points. First, a Riemannian metric is intrinsic data you put ON the manifold; it is not the same as embedding the manifold in Euclidean space, and the same M carries infinitely many inequivalent metrics. Second, 'metric' here means a positive-definite inner product field, NOT a distance function in the point-set sense — although g does induce a genuine distance d(p,q) as the infimum of lengths of curves. If you only require g to be nondegenerate (allowing a sign), you get a pseudo-Riemannian metric; the Lorentzian signature used in relativity is the famous example and is a different category.
On the upper half-plane {(x,y) : y > 0} the metric ds^2 = (dx^2 + dy^2)/y^2 makes vertical lines and semicircles meeting the x-axis at right angles into geodesics, and produces a surface of constant curvature -1: this is the Poincare half-plane model of hyperbolic geometry, built purely from a choice of g.
The same smooth manifold (an open half-plane) becomes hyperbolic space purely by the choice of metric.
'Metric' is overloaded: a Riemannian metric is an inner-product field (positive-definite, has a sign-free notion of length), whereas a metric space's metric is a distance function. The first induces the second, but they are not the same object.