Basically, the atomic bonds. A material is a strong as the weakest bond.
I'm going to go ahead and disagree with this. There are many examples of materials becoming stronger through the deliberate introduction of weak or unsatisfied bonds. One example is in metals, where additional dislocations and grain boundaries (1-D and 2-D defects, respectively, consisting of distorted and broken bonds) strengthen the material. Another example is ceramics, where voids (no bonds at all!) are used to interrupt crack propagation and again strengthen the material.
The statement isn't even true in a general sense: covalent and ionic bonds are generally stronger than metallic bonds, but metals are generally stronger than covalent and ionic crystals in tension!