NAND gates are one of the two basic logic gates (the other being NOR logic) from which any other logic gates can be built. Due to this property, NAND and NOR gates are sometimes called "universal gates". However, modern integrated circuits are not constructed exclusively from a single type of gate. Instead, EDA tools are used to convert the description of a logical circuit to a netlist of complex gates (standard cells) or transistors (full custom approach).
If the truth table for a NAND gate is examined or by applying De Morgan's Laws, it can be seen that if any of the inputs are 0, then the output will be 1. However to be an OR gate, if any input is 1, the output must also be 1. Therefore, if the inputs are inverted, any high input will trigger a high output.
An XOR gate is constructed similarly to an OR gate, except with an additional NAND gate inserted such that if both inputs are high, the inputs to the final NAND gate will also be high, and the output will be low. This effectively represents the formula: "(A NAND (A NAND B)) NAND (B NAND (A NAND B))".