There are a couple of ways to answer this question. I generally think in terms of crystal systems and lattice types. There are 7 crystal systems:

triclinic - usually not symmetrical from one side to the other, which can lead to some fairly strange shapes
monoclinic - like skewed tetragonal crystals, often forming prisms and double pyramids
orthorhombic - like tetragonal crystals except not square in cross section (when viewing the crystal on end), forming rhombic prisms or dipyramids (two pyramids stuck together)
tetragonal - similar to cubic crystals, but longer along one axis than the other, forming double pyramids and prisms
trigonal - possess a single 3-fold axis of rotation instead of the 6-fold axis of the hexagonal division
hexagonal - six-sided prisms. When you look at the crystal on-end, the cross section is a hexagon
cubic - not always cube shaped! You'll also find octahedrons (eight faces) and dodecahedrons (10 faces).
Lattices can either be primitive (only one lattice point per unit cell) or non-primitive (more than one lattice point per unit cell).

If you combine the 7 crystal systems with the 2 different types of lattices, you end up with 14 Bravais Lattices (named after Auguste Bravais who figured all this out in 1850).
Another way to answer your question is to categorize crystals by their physical/chemical properties. In this classification you have four types of crystals:

Covalent Crystals:
This is a crystal which has real chemical covalent between all of the atoms in the crystal. So really a single crystal of a covalent crystals is really just one big molecule. An example of this is a crystal like diamond or zinc sulfide. Covalent crystals can have extremely high melting points. 
 
Metallic Crystals:
Individual metal atoms sit on lattice sites while the outer electrons from these atoms are able to flow freely around the lattice. Metallic crystals normally have high melting points and densities. 
 
Ionic Crystals:
This is a crystal where the individual atoms don't have covalent bonds between them, but are held together by electrostatic forces. An example of this type of crystal is sodium chloride (NaCl). Ionic crystals are hard and have relatively high melting points. 
 
Molecular Crystals:
This is a crystal where there are recognizable molecules in the structure and the crystal is held together by non-covalent interactions like van der Waals forces or hydrogen bonding. An example of this type of crystal would be sugar. Molecular crystals tend to be soft and have lower melting points.