As the Earth moves around the Sun in its orbit, we periodically pass through debris trails left behind by comets. As these small bits of dust (many of them only slightly larger than microscopic) get swept up by our planet like a Swiffer sweeper, the particles plummet through our atmosphere toward the ground. During this plunge, they compress the air in front of them, causing it to heat up (and therefore heat up the particle as well) and the result is a glowing streak (a meteor) flying through the air. When we pass through a comet trail and experience a large number of meteors, we call this a meteor shower.
This Thursday night / Friday morning, we are set to experience the peak of the Leonid meteor shower, associated with the comet Tempel-Tuttle. While the performance of this meteor shower from year to year is somewhat inconsistent, the Leonid meteor shower is typically one of the better showers of the year. Predictions of the number of meteors per hour can vary, but seeing at least one meteor per minute is not uncommon for this shower. The particles are moving extremely fast, at roughly 72 km/s, so you'll have to have a sharp eye out for those streaks as they light up the sky!
The shower is called the Leonid meteor shower because the meteors appear to radiate from the constellation Leo. Meteors are best viewed after midnight, as the viewer is on the leading side of the Earth in its rotation (think of a car's front windshield catching more rain drops than the rear windshield), but this year the bright third-quarter moon may wash out all but the brightest meteors. However, while the meteor showers are typically best viewed after midnight, meteors can be seen all night long and so it's worth stepping outside for 5-10 minutes in the evening before the moon rises to see if you can spot them. Face east, and look up.
Happy viewing!