What are periods in physics(pendulums)?

From distance = average speed X time traveled; if the distance is the length of a wave L = Vavg T, then by definition T is the period. In other words it's just the time to travel one wavelength. When we measure L as 1 cycle, the period is the time to go through one cycle.

Don't agree with the av8r on the second part. Perhaps its a matter of interpretation for your givens.

But I read "moves through equilibrium every 1s" as taking 1 second per half cycle. So the period is actually T = 2 seconds to complete one cycle, which is to swing all the way left, come back right to neutral, swing all the way to the right, and come back to neutral.

In other words, from full left to full right is 1/2 s + 1/2 s = 1 second. But to return to the full left is another 1 second, so the period is T = 2 seconds.