Building on Bob's answer, and to your question about equations:
The basic concept to remember is DeMorgen's Theorem. Using + for OR, * for AND, and ~ for NOT,
~(a + b) = ( ~a * ~b )
~(a * b) = ( ~a + ~b )
In other words, the output of a NOR gate is equivalent to the output of an AND gate with the inputs inverted. And, vice versa: The output of a NAND gate is equivalent to the output of an OR gate with the inputs inverted.
If you move the inversions all to one side you get:
(a + b) = ~( ~a * ~b )
(a * b) = ~( ~a + ~b )
In other words, an OR gate is equivalent to a NAND gate with inverted inputs, and an AND gate is equivalent to an OR gate with inverted inputs.
The trick to realize is that you can move the "bubbles" around and implement DeMorgen's theorem with the schematic. I've heard this called "the bubble game." The idea is to figure out what function you need with just "positive logic" using ANDs and ORs. Then play the bubble game and make them all NANDs and NORs with bubbles on the inputs, then move the bubbles along the lines (two on a line cancel) to make simple NANDs and NORs. Sometimes you need an extra inverter here or there, too.
The bubble game has four rules:
1) You can change ANDs or ORs to (N)ANDs and (N)ORs with bubbles on all terminals.
2) You can "push" a bubble from the output back to the inputs, making them all inverted.
3) You can "push" bubbles from all inputs through to the output, inverting the output.
4) Two bubbles on a line cancel.
Here's an example.
It turns out if we only change the output gate we can save a step or two...