This is a relatively nice shower to watch. There are usually 20 - 30 meteors per hour, so you'll see one every few minutes. Sometimes -- though rarely -- it can peak at much higher rates, but I don't think anything like that is predicted this year.
Watching a meteor shower is actually pretty easy. All you need is a big view of the open, dark sky -- the fewer trees, buildings and lights the better -- and something comfortable to relax on like a beach chair (the kind that opens up so you can lie on it). The best time to watch is after midnight; that's when the Earth is facing into the oncoming bits of gravel and ice, and you see more meteors (like seeing more bugs hitting your car's front windshield than the rear one). The Moon is not quite full, and should be low in the sky after local midnight. Not ideal observing conditions, but not too terrible.
If you want details, I wrote an article on 12 things you need to watch a meteor shower. It was written for the 2007 Perseids in August, but it still applies. Just dress warmly!
Also, the folks at the UK's Sky at Night magazine put together a nice informational video about the Leonids and meteor showers (it's from last year, but still relevant):
That's it! Happy meteor hunting!
Leonids over Uluru image courtesy Vic and Jen Winter at ICSTARS.
- Watch the Leonids!
- Will the Leonids roar in 2009?
- 12 things you need to watch the Perseid meteors Sunday night