Some meteors from the Perseids should be visible for most of the month but the peak will be on the morning of August 12, when we can expect nearly 60 meteors per hour if conditions are right. The sky will be especially dark at this time because the moon will be three days past last quarter and won't rise until 1 a.m. on the morning of the 12th, so the meteors should be clearly visible in the dark sky.
As to the moon, the first full moon is on August 1; it's called the Grain, Green Corn or Sturgeon Moon (hey, I don't make this stuff up). The second full moon occurs on the 31st, and because it's the second full moon in a calendar month it's called a Blue Moon (wasn't there a song about that? - yes, there was so let's listen: