first fundamental form
Suppose you are an ant living on a curved surface — a sphere, a cylinder, a saddle — and you can only crawl along it; you cannot see the surrounding 3-D space. What can you still measure? You can measure distances along paths you walk, angles between directions, and areas of patches. The first fundamental form is the complete record of all such intrinsic measurements: it is the surface's own ruler, inherited from the way it sits in space, but usable entirely from within.
Concretely, take a surface described by a parametrization r(u, v). A small step (du, dv) in the parameters produces a real displacement dr = r_u du + r_v dv on the surface, and the squared length of that displacement is ds^2 = E du^2 + 2F du dv + G dv^2, where E = r_u dot r_u, F = r_u dot r_v, and G = r_v dot r_v. This expression is the first fundamental form, often written I. The three coefficients E, F, G are exactly the components of the surface metric tensor g_{ij} in the (u, v) coordinates: E and G are the diagonal entries, F the off-diagonal one, with F = 0 precisely when the coordinate curves cross at right angles. From I alone you compute arc length (integrate sqrt of I), angles (via the dot product it encodes), and surface area (integrate sqrt(EG - F^2) du dv).
The first fundamental form is the foundation of the intrinsic geometry of surfaces. Everything an inhabitant can determine without leaving the surface — geodesics (shortest paths), the Gaussian curvature (by Gauss's celebrated Theorema Egregium), parallel transport, the angle sums of triangles — is computable from I and its derivatives. This is the conceptual bridge to Riemannian geometry and general relativity: spacetime, too, is known only through its metric, with no surrounding higher space to look in from. The honest distinction to keep straight: the first fundamental form captures only intrinsic data (how the surface measures itself); how the surface bends within the surrounding space is extrinsic information carried by the second fundamental form, which I knows nothing about.
A flat sheet of paper and a rolled-up cylinder have the SAME first fundamental form (rolling does not stretch the paper), so an ant cannot tell them apart by measuring distances and angles on the surface. This is why you can roll paper into a cylinder without tearing or wrinkling — and why a flat map of a sphere is impossible without distortion, since the sphere's I genuinely differs from the plane's.
Paper and cylinder share the same I — equal intrinsic geometry — even though they bend differently in space.
The first fundamental form encodes only intrinsic geometry; a flat plane and a bent cylinder share it. Whether a surface bulges or dimples into the ambient space is extrinsic and lives in the second fundamental form — yet, remarkably, Gaussian curvature turns out to be intrinsic, computable from I alone.