Stability is a highly overloaded word in mathematics.
property of a morphism can be stable under pullbacks
the equations (for example differential equations) may depend on parameters like initial condition. If the parameters change a little bit, the solution may change more or less radically. Thus one can talk about stability of solutions of differential equations under perturbations. This is also related to the notion of stable equilibria in physics and engineering.See wikipedia stability theory
the notion of stability of particles and bound states in physics: stability under decay
in model theory there is another theory of stability including a more recent geometric stability theory
in geometry, specifically in the construction of moduli spaces there is a notion of stable and semistable objects (e.g. Mumford stability in geometric invariant theory); they can be taken with respect to some stability data; more recently those are important in physical applications and abstract versions appeared in the context of triangulated categories, most notably Bridgeland stability conditions.
See also stable vector bundle