the curvature of a curve
Everyone has an intuition that a hairpin bend is 'sharper' than a gentle sweep, and a tight circle is 'more curved' than a wide one. Curvature is the precise number that turns this feeling into mathematics. It measures how fast a curve is changing direction as you travel along it: a straight line, which never changes direction, has curvature zero everywhere, while a small tight circle, whose direction swings around quickly, has large curvature.
The clean definition uses the unit-speed (arc-length) parametrization. The unit tangent T tells you the direction of travel; as you move a small distance ds, T rotates a little, and the curvature kappa is the rate of that rotation: kappa = |dT/ds| = |T'(s)|, the magnitude of how fast the direction vector is turning per unit length travelled. For a general (not unit-speed) parametrization there is a working formula: in the plane, kappa = |x' y'' - y' x''| / (x'^2 + y'^2)^(3/2), and in space, kappa = |r' x r''| / |r'|^3. A beautiful fact ties curvature to circles: at each point a curve has a best-fitting circle, the osculating circle, and the curvature is simply 1 over that circle's radius, kappa = 1/R. So a circle of radius R has constant curvature 1/R — a big circle (large R) is gently curved (small kappa), a tiny circle is sharply curved.
Curvature is fundamental because, together with torsion, it determines a space curve completely up to rigid motion — that is the fundamental theorem of curves. It also shows up physically: the centripetal acceleration needed to follow a path at speed v is v^2 times kappa, which is why tight turns at speed throw you sideways hard. Two honest cautions are worth stating loudly. First, curvature is a precise quantity, not a vague impression of 'how curvy it looks' — it has a formula and a value at every point. Second, the curvature of a curve (how the curve bends in the space around it, an extrinsic notion) is a different thing from the Gaussian curvature of a surface (an intrinsic notion, the subject of the Theorema Egregium); never conflate the bending of a wire with the curvature of a sheet.
A circle of radius 2 has curvature kappa = 1/2 at every point; a circle of radius 1/3 has curvature 3 everywhere — three times as sharply curved. A straight line has kappa = 0. For the parabola y = x^2, using kappa = |y''| / (1 + y'^2)^(3/2) with y' = 2x, y'' = 2, the curvature at the vertex x = 0 is 2/(1)^(3/2) = 2, and it decreases toward 0 far out along the arms.
kappa = 1/R: small radius means large curvature; a straight line has kappa = 0.
Do not confuse a curve's curvature (extrinsic — how a 1D curve bends in surrounding space) with a surface's Gaussian curvature (intrinsic — the subject of the Theorema Egregium). They are genuinely different concepts that happen to share the word 'curvature'.