Depending on the size of the alloying element, a substitutional solid solution or an interstitial solid solution can form. In both cases, atoms are visualised as rigid spheres where the overall crystal structure is essentially unchanged. The rationale of crystal geometry to atom solubility prediction is summarized in the Hume-Rothery rules and Pauling's rules. 
Substitutional solid solution strengthening occurs when the solute atom is large enough that it can replace solvent atoms in their lattice positions. Some alloying elements are only soluble in small amounts, whereas some solvent and solute pairs form a solution over the whole range of binary compositions. Generally, higher solubility is seen when solvent and solute atoms are similar in atomic size (15% according to the Hume-Rothery rules) and adopt the same crystal structure in their pure form. Examples of completely miscible binary systems are Cu-Ni and the Ag-Au face-centered cubic (FCC) binary systems, and the Mo-W body-centered cubic (BCC) binary system.
Interstitial solid solutions form when the solute atom is small enough (radii up to 57% the radii of the parent atoms) to fit at interstitial sites between the solvent atoms. The atoms crowd into the interstitial sites, causing the bonds of the solvent atoms to compress and thus deform (this rationale can be explained with Pauling's rules). Elements commonly used to form interstitial solid solutions include H, Li, Na, N, C, and O. Carbon in iron (steel) is one example of interstitial solid solution.