How does alternating current provide energy? In my head, direct current makes complete sense; the electrons carry energy around the circuit to something being powered losing its potential and then return to the battery or whatnot to have their potential raised again.
This is probably wrong, so I would like an explanation of how a direct current actually transfers energy and also how an alternating current does (I have zero intuition for this).
 A: Imagine a current going through a resistor and generating heat.
Does it matter which way the current goes?  No, it doesn't; you get heat either way.  So reversing the current many times a second, as AC, still generates heat.
Sometimes people get tangled up because they somehow think that electrons are "used up" in electric circuits.  But they're not.  The power source gives them energy and sends them through the circuit, where the energy is lost.  That still works if the power source sends them through the other way.
A: Here is an analogy which might help.
Imagine you are cutting a board with a saw. You push the saw through the board, and it cuts. Then you lift the saw up out of the board, pull it back, set it back into the board, and push it through again, always in the same direction. Eventually, the board is cut. Or...
You  can push the saw through the board, and then pull it back while in the board, push it forward again, then pull it back, and so forth. As long as the saw teeth are shaped to cut on either stroke, you'll cut the board by going back and forth instead of in one direction only. 
Running the saw one way only through the board is like DC power. Pulling the saw back and forth is like AC power. 
