the first variation formula
Stretch a rubber band between two pins, then nudge it sideways a little and ask: did its length go up or down, and how fast? The answer to that question — the rate of change of length (or area, or energy) as you wiggle the object — is the first variation. In ordinary calculus you find a lowest point by setting a derivative to zero; here the 'variable' is an entire curve or surface, and the first variation is the derivative of a length-, area-, or energy-functional with respect to a one-parameter family of nearby competitors. Setting it to zero is exactly the condition for the object to be a critical point: a geodesic, a minimal surface, a harmonic map.
Concretely, take a smooth curve gamma in a Riemannian manifold (M, g) and a variation gamma_s(t) with variation field V(t) = d/ds gamma_s at s=0. For the energy E(gamma) = (1/2) integral |gamma'|^2 dt the first variation is dE/ds at s=0 = -integral over the curve of g(V, nabla_{gamma'} gamma') dt + [boundary terms g(V, gamma')]. Notice nabla_{gamma'} gamma' is the acceleration; the integral vanishes for every V fixing the endpoints exactly when nabla_{gamma'} gamma' = 0, which is the geodesic equation. For arc length the same computation, after reparametrizing by arc length, gives dL/ds = -integral g(V, kappa) ds with kappa the curvature vector, so curves of zero geodesic curvature are the critical points. For a submanifold Sigma the first variation of area is d(Area)/ds = -integral over Sigma of g(V, H) dA, where H is the mean curvature vector; critical points have H = 0.
The formula is the engine that turns geometry into analysis: a geometric object 'is straight/minimal/harmonic' becomes an Euler-Lagrange equation you can attack with PDE methods. A standard caution: a critical point need not be a minimum. The first variation only finds stationary configurations; a geodesic can be a saddle of the energy (think of the long way around a sphere), and only the second variation can tell minima from saddles. Also, the boundary term is not optional — it is what enforces fixed-endpoint or free-boundary conditions, and dropping it silently changes the problem.
On the round sphere both the short arc and the long arc of a great circle joining two non-antipodal points satisfy nabla_{gamma'} gamma' = 0, so both are geodesics and both make the first variation of energy vanish — yet only the short one minimizes length.
Vanishing first variation = critical, not minimal: the long great-circle arc is a non-minimizing geodesic.
Energy and arc length have the same critical points only up to reparametrization: minimizing energy automatically gives constant-speed parametrization, while arc length is reparametrization-invariant, so energy is usually the friendlier functional to differentiate.