What are the two main categories of logic circuits?

The following are all common digital electronic logic gate types:

AND

OR

NAND

NOR

XOR

NOT

There 'truth tables' and therefore their corresponding functions can be found on wikipedia under articles entitled with their respective names.

NAND is a very common gate in commercial digital electronics as all other gates can be imitated by connected the correct number of NAND gates in the correct way. These can then be placed on one IC (Integrated Circuit) chip which reduces cost massively.