Most meteor showers vary from year to year, but the Leonids are particularly capricious. Many years they chug along producing just 5 or 10 meteors visible per hour. But at the Leonids' historical greatest, in 1833, meteors were seen to fall "like snowflakes in a blizzard," with estimated rates of several dozen per second!
This year is expected to be better than average. The "traditional," most reliable part of the shower should peak around 4 a.m. EST (1 a.m. PST) on the morning of Tuesday, November 17th. You might see 20 or 30 meteors per hour under ideal dark-sky conditions. (Remember, if you want to stay up late instead of getting up early, you'll be staying up Monday night. It's easy to get the date wrong for events that happen after midnight!)
A second, briefer, but very intense outburst is expected about 12 hours later — during the early-morning hours of November 18th in Asia. (See "Will the Leonids Roar Again?".) There's only an off-chance that some activity from that burst will still be going on by the time the Earth turns halfway around and the Leonids become visible in the Americas on the morning of the 18th.
But if the sky is clear, why not go out again that morning — and also before the predicted peak, on the morning of the 16th? The Leonids have surprised the theorists before, and they surely will again.
Wherever you are, no Leonids will be visible before the shower's radiant point (in Leo) rises around local midnight. And peaks and bursts aside, the number of visible meteors increases steadily from radiant-rise until Leo is highest, just as the sky is starting to get light.
Be sure to bundle up warmly; meteor-watching is always colder than you expect. Ideal meteor-watching equipment is a comfortable lounge chair, a warm sleeping bag, and a pillow. If you live in a city or suburb, consider traveling to a dark location far from city skyglow. In any case, find a spot where no lights glare directly into your eyes.
The direction to watch is wherever your sky is darkest. Notice the meteors' flight paths; only those streaking away from the direction to the constellation Leo are Leonids.
Another, less-known meteor shower is going on simultaneously — the Taurids. They're sparse but tend to be very bright. If you see a slow, bright meteor heading away from the direction to Taurus, that's a Taurid.
And you're bound to see a few sporadics that aren't associated with any major shower.