a surface parametrization
A flat map of a curved world is the everyday version of this idea. A globe is awkward to carry, so we flatten regions of it onto paper, with grid lines of latitude and longitude. Each spot on the map corresponds to a spot on the globe, and the grid lets you name places by two numbers. A surface parametrization does exactly this for any curved surface: it spreads a flat sheet of (u, v) coordinates over a patch of the surface, so that every point of the patch gets an address (u, v).
Concretely, a parametrization of a patch is a smooth map x(u, v) from a flat region of the (u, v)-plane into space, landing on the surface, that is one-to-one and regular (its partials x_u and x_v are independent everywhere, so x_u x x_v is never zero). The two partials are the velocity vectors you feel if you move along the surface keeping v fixed and letting u run (that gives x_u), or keeping u fixed and letting v run (that gives x_v). The curves traced by holding one coordinate fixed are the coordinate curves; together they form the curvilinear grid drawn on the surface. To find points on the surface you simply feed in (u, v); to do geometry there you differentiate x with respect to u and v.
Parametrizations are the practical engine of the whole subject: tangent planes, normals, lengths, areas, and curvatures are all computed by differentiating x(u, v). Two honest cautions. First, a single parametrization rarely covers a whole closed surface — latitude-longitude breaks down at the poles of a globe, where the grid bunches up — so we use several overlapping parametrizations and check they agree on overlaps. Second, the (u, v) coordinates are a CHOICE, like the choice of map projection; the surface itself does not know about them, and any genuine geometric fact must come out the same no matter which parametrization you used.
A sphere of radius R has the latitude-longitude parametrization x(u, v) = (R cos u cos v, R cos u sin v, R sin u), where u is latitude (from -pi/2 to pi/2) and v is longitude (0 to 2pi). Plugging in u = 0, v = 0 gives the point (R, 0, 0) on the equator. The partials x_u and x_v point 'north' and 'east' along the surface — independent everywhere except at the poles u = +-pi/2, where x_v collapses to zero, so the parametrization fails exactly there (just as a flat map distorts hopelessly at the poles).
Latitude-longitude parametrizes the sphere everywhere except the two poles.
A parametrization is a tool, not the surface. The same patch admits infinitely many parametrizations; quantities that depend on the choice (like the raw coordinates) are bookkeeping, while quantities that come out the same in every parametrization (lengths, areas, curvatures) are the real geometry.