Late tonight (Nov 16) and into the early hours of the tomorrow morning is the best time to see the peak of this year's Leonid meteor shower. The name comes from the constellation from which the Shooting Stars, or meteors, appear; in this case, Leo, the Lion.
So, from 10:30pm(ish) you should begin to see the odd streak of light across the sky appearing anywhere in the sky zooming in an east to west direction.
With clear, dark skies and good horizons the latest estimates suggest up to 150 shooting stars (that's right!) per hour may appear during the maximum which is just after midnight as the 16th becomes the 17th.
Unfortunately the gibbous Moon will be a natural light-polluting nuisance by washing out the light of the fainter meteors over the entire time, but you'll still be able to catch the brighter ones.