I am having trouble understanding how electron flows in an AC current. In AC current I understand the potential difference is relative to the neutral lead, the electrons are vibrating. I am trying to imagine an alternating current with a slow cycle, say one cycle every 10 seconds, do the electrons flow towards the neutral lead for half the cycle then back towards the live lead for the other half? Is that correct? Is there an analogy to help me understand?

good Question! I am asking myself the same thing for over an year and I thought at the answer as you did .. maybe someone will answer to understand for sure how things work!

The polarities of the conductors determine the direction of the current. Remember that electrons are negative charge carriers, so they go towards the more positive potential. In an AC circuit, electrons go both ways, towards hot from neutral and then towards neutral from hot, all influenced by the polarity of the applied voltage.

Historically, electrified trains ran on AC as low as 10 Hz.

Let's say in the case of a incandescent light bulb hooked up to a 117VAC 60Hz circuit. The light bulb filament gives off light when a a current flows through it. So when an alternating current flows through it, it gives off the most light (the most current) at the maximum and minimum +V and -V and zero current at 0V at half way and the end of each cycle. Correct?

You know that AC voltage is a sine wave right? Well, just pick a point on that sine wave. If it's positive, at that instantaneous time that voltage is the same as DC. If it's above the middle line (ground) then it is a positive voltage, and below it is a negative voltage (like flipping the leads from a power supply). And yes, the maximum power is produced at the peaks of the sine wave. By dividing AC by the square root of 2, you essentially convert it to it's DC form, which you use for finding power in a circuit.