Are there different categories or types of tornadoes?

Yes. The Enhanced Fujita scale has six intensity categories ranging from EF0 for the weakest tornadoes to EF5 for the strongest. This identifies different levels of strength rather than actual types of tornado. As far as actual types, there are classic supercell tornadoes, which form from a larger mesocyclone, and weaker landspouts, which form as a result of processes beneath a thunderstorm.