This Sunday night (into early Monday morning) and the following Monday night (into early Tuesday morning) will be the best opportunities to see the annual Perseid Meteor Shower.
What is a meteor shower?
When the Earth in its orbit passes through dust left over from a comet, some of that dust enters the atmosphere and burns up, creating a streak of light called a meteor (or shooting star). The leftover dust stays very close to its parent comet’s orbit, so the Earth passes through it at the same time every year.
In the case of the Perseid Meteor Shower, the comet that produced the dust that can become meteors is Comet Swift-Tuttle. Swift-Tuttle orbits the sun once every 133.3 years, and its last close approach to the sun was in December 1992 (so it won’t be back in our neighborhood until 2126).
What can you expect to see?
From the city, probably not very many meteors, but you never know. One year I happened to be in San Francisco and watched the sky from a friend’s back patio, and I saw one about every 5-10 minutes or so. You shouldn’t expect to see blazing streaks of light (like the Millennium Falcon jumping to light speed below), but if you are willing to sit outside for a few hours, you should see a few brief streaks of light across the sky.
You won’t need any special equipment, just a chair. And you don’t need to look in any specific direction. They can appear anywhere in the sky. If you could freeze every meteor you saw and trace their paths back, they would all appear to start from a point in the constellation Perseus, which is how the shower got its name. But you don’t even need to know where Perseus is in order to see meteors (but if you are curious, it is in the northeast and rises higher as the night turns into morning).
The darker your sky, the more meteors you will see. So if you can get away from the city, do so. The Adler Planetarium is sponsoring a star party on Monday night at Cantigny Park in Wheaton, for example. Even if you can’t escape the city lights, patience is the key. And the later you stay up, the better the viewing will get; most meteors appear after midnight when the Earth rotates into the direction of the comet dust. Imagine driving down a highway; more bugs hit your front windshield than your side windows. After midnight, your view becomes more like the view through the front.
Good luck, and here is an orientation video from NASA/JPL.