The three groups of rocks are classified by?

Igneous: these rocks are volcanic in nature. They are split into two categories in where they cool/crystallize: intrusive (internal) and extrusive (external). Intrusive volcanic rocks cool slowly, so they contain (usually) large crystals, like granites. Extrusive rocks are quite the opposite, in which they cool very quickly. This is where you get your obsidian glass and basalts. 
Sedimentary: these rocks are depositional and categorized by the kind of sediment it is made up of. 
Metamorphic: this can be the offspring rock of either a morphosed igneous or sedimentary rock. This metamorphism is accomplished by heat and/or pressure, and during a great length of time.