What is a mineral? Any naturally formed solid,
chemical substance having a definite chemical composition
and a characteristic crystal structure.
Here's an example: Pyrite (FeS2), also known as "Fool's gold."
http://www.argentumauctions.com/catalogue/carnegie/2000/photos.html
Crystal structure means a nice, regular lattice where
each atom (Fe or S, in this case) has a specific place
it belongs.
http://www.chem.lsu.edu/htdocs/people/sfwatkins/ch4570/lattices/lattice.html
Other famous minerals you have probably heard of include
quartz, calcite, garnet, and olivine.
You can find other minerals
here
(gemstones are minerals; some are more pure and more
rare than others).