Stiffness is a matrix that relates the stresses to the strains in a solid. For isotropic materials--noncrystalline materials that have rotationally-symmetric properties--the stiffness matrix is very simple, consisting of only two independent terms. Only two terms are necessary since for isotropic materials, the shear modulus, Young's modulus, and Poisson's ratio are related by the formula G = E/2(1+v) where G is shear modulus, E is Young's modulus, and v is Poisson's ratio. Strictly speaking, those constants are only defined for isotropic materials.
In general, the stiffness matrix is a 6x6 symmetric matrix. The relationship between stress and strain can be written σ = Cε, where σ is a column vector that contains both normal stresses and shear stresses, and ε is a column vector that contains both uniaxial strains and shear strains*. The matrix equation can be written compactly as σi = ΣCijεj, where i and j of 1, 2, and 3 are for normal stresses and strains and i and j of 4, 5, and 6 are for shear stresses and strains.
* The adjectives uniaxial and normal can be removed--they are just there to differentiate from shear strains and stresses.
The greater the symmetry of a crystal (i.e. the more isotropic), the fewer the number of independent coefficients Cij. For example, crystals like silicon with cubic symmetry have only three independent coefficients. In cubic crystals C11 = C22 = C33, C12 = C13 = C23 (and C is symmetric), C44 = C55 = C66, and all other elements are 0.
The inverse of the stiffness is called compliance.