A problem is well-conditioned when small changes in its input produce only small changes in its exact output. It is ill-conditioned when tiny input perturbations can cause much larger output changes.

For a scalar function $y=f(x)$, local sensitivity can often be related to the derivative. A large $|f'(x)|$ means a small absolute perturbation in $x$ can produce a larger perturbation in $y$.

A dimensionless condition number compares relative changes. Informally,

$$\kappa\approx\frac{\text{relative output change}}{\text{relative input change}}.$$

Large $\kappa$ means the problem amplifies uncertainty.

Conditioning is a property of the mathematical problem, not of the algorithm used to solve it. No algorithm can reliably recover information that the problem itself makes extremely sensitive to input or measurement errors.