the elliptic type
Picture a soap film stretched across a bent wire loop, or the steady temperature in a metal plate whose edge is held at fixed values, or the electrostatic potential between charged plates. In each case nothing is changing in time — everything has settled into equilibrium, and the value at every interior point is a kind of gentle average of its neighbours. Equations describing this settled, equilibrium world are the elliptic type, the calmest and smoothest of the three families.
Formally, a second-order linear PDE A u_xx + 2B u_xy + C u_yy + (lower order) = 0 is elliptic at a point when its discriminant B^2 - AC < 0 there. The name comes from the associated quadratic form A p^2 + 2B p q + C q^2, whose level curves are ellipses in that case. The defining archetype is Laplace's equation u_xx + u_yy = 0 (here A=C=1, B=0, so B^2 - AC = -1 < 0). The hallmark consequence is that there are NO real characteristic curves: the equation has no preferred directions along which information streams, which is exactly why elliptic solutions are so featureless and smooth — in fact solutions of Laplace's equation are real-analytic, infinitely differentiable, even where the boundary data is rough.
What kind of data is right for an elliptic equation? Boundary data all the way around a closed region — prescribe u on the whole boundary (a Dirichlet problem) or its normal derivative (a Neumann problem), and the interior is determined. This is a boundary-value problem, not an initial-value problem: there is no time to march forward in. A crucial honest warning lives here — trying to impose Cauchy (initial) data on an open piece of curve and march, as you would for a wave, is ILL-POSED for an elliptic equation (Hadamard's example): the tiniest wiggle in the data makes the solution explode. The type really does dictate the data.
The Dirichlet problem for Laplace's equation on a disk — fix the temperature around the rim, ask for the temperature inside — is the model elliptic problem; it has exactly one smooth answer (given by the Poisson integral), and the answer at the centre is just the average of the rim values.
Elliptic = equilibrium: data on the whole boundary, no time direction, a single very smooth solution.
A common misconception is that elliptic equations are the easy ones because they look so tame. The tameness is real, but it is also why Cauchy data is forbidden: an elliptic equation has no domain of dependence to march along, so an initial-value formulation is ill-posed.