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How do iron and titanium impurities give sapphire its blue color?

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A rough, translucent yellow crystal of elemental sulfur displayed against a solid black background. — crystal A crystal form of elemental sulfur. Sulfur has an atomic weight of about 32.06 atomic mass units (amu), which means that its molar mass is 32.06 grams.

Pure aluminum oxide, also known as corundum, is naturally colorless. While individual impurities of iron or titanium do not produce a deep blue color on their own, the presence of both together creates the magnificent blue hue of sapphire.12

This coloration is caused by a process called charge transfer. When light energy is absorbed, an electron is temporarily transferred between the iron and titanium ions, resulting in a valence change:
$F e^{2 +} + T i^{4 +} \to F e^{3 +} + T i^{3 +}$
This specific transition requires energy that corresponds to the absorption of light in the yellow region of the spectrum. Because the crystal absorbs yellow light, the complementary color blue is transmitted, giving the sapphire its characteristic appearance.2