Alumina is a white, transparent ceramic material. Its chemical composition is Al2O3. This means it has the same chemical composition as sapphire or ruby. Unlike these precious gems, however, alumina is colorless, not having the metal impurities that give the gems their color. The natural form is called corundum. The name alumina usually means the synthetic form.

Alumina appears in both a polycrystalline and a monocrystalline form, usually hexagonal. This form is called alpha-alumina. The other from, cubic gamma-alumina is less stable. Amorphous alumina does not exist.

Alpha-alumina is an extremely hard material, having a hardness of 9 on the Mohs scale. This makes it well suited for many technical applications, as it's a lot cheaper than other hard materials like diamond.

Furthermore, alpha-alumina has excellent thermal properties. There are many ceramics with high melting points, and alpha-alumina's melting point of 2300 K is not really special. However, it also has a decent thermal conductivity of 30 Watt per meter per Kelvin at room temperature. Compare this to a measly 0.8 W/m K for quartz and you see why alpha-alumina is popular for making things that must withstand extreme heat. Its ability to withstand temperature changes is further augmented by a decent specific heat of 77 Joule per kilogram per Kelvin. These properties are employed for instance in high-powered plasma lamps and ore smelters.

Polycrystalline gamma-alumina can be made in a form that has many holes in it. These holes greatly enhance the surface of the material, making gamma-alumina a popular material to use in chemical catalysts.

Alumina is not known to be very hazardous, although inhaling fine alumina dust is probably not a good idea. This makes it a far more desirable material to use than the related beryllia, which has better thermal properties, but is extremely toxic.

Sources:
The trusty Handbook of Chemistry and Physics