What is Fajans rule?
Fajans' rules (note the difference) were formulated in 1923 by Kazimierz Fajans.

They are a method for predicting ionic vs. covalent that predates electronegativity (by three decades) and make use of ionic and atomic radius data that was becoming available through x-ray crystallography. 

To use Fajans' Rules, assume your binary compound is ionic and identify the potential cation and anion.

By Fajans' Rules, compounds are more likely to be ionic if: there is a small positive charge on the cation, the cation is large, and the anion is small. For example, $\ce{NaCl}$ is correctly predicted to be ionic since $\ce{Na+}$ is a larger ion with a low charge and $\ce{Cl-}$ is a smaller anion.

Compounds are more likely to be covalent if: there would be a large positive charge on the cation, the cation would be small, and the anion would be large. For example, $\ce{AlI3}$ is correctly predicted to be covalent since it would have a small cation with a high charge and a large anion. 

Note that Fajans' Rules have been largely displaced by Pauling's approach using electronegtivites. However, the remnants of Fajans' Rules are found in Hard-Soft Acid-Base Theory, which predicts bonding properties based on polarizability (which is based on size and charge). Binary compounds having a soft acid and/or a soft base are often covalent.

$$\begin{array}{|c|c|c|c|}\hline
\mathrm{compound} &amp; \mathrm{Fajans} &amp; \mathrm{Pauling} &amp; \mathrm{HSAB}\\ \hline
\ce{NaCl} &amp; \mathrm{low + charge \\ larger\ cation \\ smaller\ anion \\ionic} &amp; \mathrm{3.16-0.93 = 2.19\\ ionic} &amp; \mathrm{hard\ acid \\ borderline\ base \\ ionic}  \\ \hline
\ce{AlI3} &amp; \mathrm{high + charge\\ smaller\ cation \\ larger anion \\covalent} &amp; \mathrm{2.66 - 1.61 = 1.05 \\covalent} &amp; \mathrm{hard\ acid\\ soft\ base \\covalent} \\ \hline
\end{array}$$