uniform ellipticity
Being elliptic at every single point is not quite enough to run the theory. If the medium becomes infinitely conductive in some direction at one spot, or grinds to a near-standstill at another, the operator can degenerate and the nice estimates fall apart. Uniform ellipticity is the quantitative promise that the ellipticity never decays and never blows up anywhere in the domain — there is a fixed floor and a fixed ceiling that hold everywhere at once.
Precisely: the operator with principal matrix A(x) is uniformly elliptic on a domain if there are two positive constants lambda and Lambda (the ellipticity constants, with lambda not greater than Lambda) such that for every point x and every direction vector xi, lambda |xi|^2 is not greater than sum over i,j of a_ij(x) xi_i xi_j is not greater than Lambda |xi|^2. The lower bound lambda greater than zero is the heart of it: it guarantees the symbol stays bounded away from zero in every direction, uniformly across the whole domain, so the operator is never even momentarily non-elliptic. The ratio Lambda over lambda measures the worst-case anisotropy of the medium. For the Laplacian, lambda = Lambda = 1.
Almost every theorem in elliptic theory quotes its constants in terms of lambda, Lambda, the dimension, and the domain — coercivity in Lax-Milgram, the Schauder and Calderon-Zygmund estimates, the De Giorgi-Nash-Moser bounds, the Harnack constant. When lambda is allowed to touch zero somewhere the equation is called degenerate elliptic, a much harder and richer world (the p-Laplacian and the porous-medium equation live there). Uniform ellipticity is the comfortable regime where the classical machinery runs without caveats.
The operator - ((1 + sin^2 x) u')' has coefficient a(x) = 1 + sin^2 x, which ranges between 1 and 2. Take lambda = 1 and Lambda = 2: then 1 times |xi|^2 is not greater than a(x) xi^2 is not greater than 2 times |xi|^2 for all x. It is uniformly elliptic. But - (x^2 u')' on the interval (0,1) is not — its coefficient x^2 sinks to zero at the left endpoint, so no positive lambda works there and the operator is degenerate.
A coefficient trapped between two positive numbers is uniformly elliptic; one that touches zero is degenerate.
Uniform ellipticity says nothing about how smooth the coefficients are — it is purely a two-sided bound on size, not on regularity. Coefficients can be uniformly elliptic and still merely measurable, with no continuity at all; that wild-coefficient case is precisely what De Giorgi, Nash, and Moser conquered.