The important quality of NAND and NOR gates is they can be used as inverters. Send the same value to both inputs and the output will be the inverse. Without an inverter you can't really get anywhere.

There are 8 possible truth tables for two inputs, so we have 8 possible gates. But two of them are obviously useless, the one which yield true for any input and one which yield false for any input. So we have 6 useful gates.

These are AND, OR, XOR, NAND, NOR and XNOR.

Of these, only NAND and NOR can be used as inverters.

Given an inverter and any of the gates you can build any other. E.g AND with an inverted output is NAND, NAND with inverted inputs are OR, OR with inverted output is NOR. XOR can be build by combining NAND and OR.