well-posed problem
When you set up a PDE together with its boundary and initial conditions, you would like to know that the problem you wrote down is a sensible model of reality before you spend effort solving it. Jacques Hadamard distilled what sensible means into three requirements, and a problem meeting all three is called well-posed. The list is short but it is the quality-control checklist of all of applied mathematics.
A problem is well-posed if: first, a solution exists (your equation and data are not contradictory); second, the solution is unique (the data pin down exactly one answer, not a family); and third, the solution depends continuously on the data (a small change in the boundary or initial values produces only a small change in the solution). The third condition, stability, is the subtle and crucial one. A problem can have a unique solution that nonetheless swings wildly when the data is nudged by a hair — and since real data always carry measurement error, such an ill-posed problem is useless for prediction even when a solution technically exists. The classic example is the backward heat equation: a unique solution exists in principle, but the tiniest noise in the present temperature explodes when you try to reconstruct the past.
Well-posedness is what ties a PDE to the correct kind of data: elliptic equations are well-posed as boundary-value problems but ill-posed as initial-value problems; hyperbolic and parabolic equations are well-posed as initial-(boundary-)value problems. Matching the equation type to its proper data is precisely the art of posing a problem correctly. Importantly, ill-posed does not mean unimportant — many real inverse problems (medical imaging, deblurring, geophysics) are inherently ill-posed, and a whole field (regularization) exists to tame them honestly rather than pretend the instability is not there.
The Dirichlet problem for Laplace's equation on a disk (prescribe u on the rim, solve inside) is well-posed: one solution, depending continuously on the boundary data. The same Laplace equation posed as an initial-value problem (Cauchy data on a line) is ill-posed — Hadamard's own counterexample blows up.
The same equation can be well-posed or ill-posed depending entirely on which data you supply.
Existence and uniqueness are not enough — the often-forgotten third condition, continuous dependence (stability), is what separates a usable model from a meaningless one when the data carry any error at all, which they always do.