the metric tensor
How far apart are two nearby points, and how much time elapses between two nearby events? On a flat sheet of paper Pythagoras answers with ds^2 = dx^2 + dy^2, but on the curved surface of the Earth, or in the warped spacetime near a star, the recipe changes from place to place. The metric tensor is the field that stores that recipe: at every point it tells you how to convert coordinate differences into real, physical distances and times. It is the single mathematical object that carries all the geometry.
Written g_ab (older texts write g_mu nu), it is a symmetric rank-two tensor with, in four-dimensional spacetime, ten independent components. It defines the line element ds^2 = g_ab dx^a dx^b, a sum over all pairs of coordinate directions. In a small enough patch it can always be brought to the flat Minkowski form eta_ab = diag(-1, 1, 1, 1), which is the equivalence principle in the language of geometry. The metric also raises and lowers indices, turning a contravariant vector V^a into its covariant partner V_a = g_ab V^b, and its inverse g^ab satisfies g^ab g_bc = delta^a_c.
In general relativity the metric is the gravitational field. Everything else, the paths of free-falling bodies, the curvature, the tides, is built from g_ab and its derivatives, and Einstein's equations are ultimately ten coupled equations for its ten components. A crucial caveat: coordinates are just arbitrary labels with no physical meaning of their own. The same physical geometry can be written in endlessly many coordinate systems; it is the metric, not the coordinates, that carries the physics, and quantities you can measure must be built so they do not depend on the labelling.
On a sphere of radius a, the surface metric is ds^2 = a^2 (dtheta^2 + sin^2(theta) dphi^2). The factor sin^2(theta) is why lines of longitude, an equal number of degrees apart, crowd together near the poles.
The metric's components vary with position; that variation is curvature.
A position-dependent metric does not by itself mean the space is curved. Flat space in polar or spherical coordinates has a non-constant metric too. Genuine curvature lives in the Riemann tensor built from the metric's second derivatives, not in the mere appearance of the components.