Take a look at the stress-strain curve for most structural steels. Stress and strain are linearly related through E (Young’s Modulus, Modulus of Elasticity) until they reach their yield point or shortly before they reach their .2% offset yield strength. Any two functions which relate and can be graphed linearly (a straight line) are acting linearly. Otherwise, they are nonlinear, and can take many forms. Certainly, when we are working in the plastic range, loads, stresses and deformations, are no longer related in a simple linear fashion. A similar case can be made for other materials, and for more complex structural elements.
There are two major types of nonlinearity:
1. Material nonlinearity as dhengr outlined above
2. Geometric nonlinearity
In general, linear approximations to problems are a good representation to actual physics when displacements, strains, temperature gradients, rotations etc are "small". (but there are exceptions)
A few examples of geometric nonlinearity: buckling of shells, stay cables of bridges, shallow arches with vertical loads, etc.
