There are a number of phenomena that "go flash in the night", with shooting stars, formally known as meteors, being among the more common. The Earth actually gets hit by some 300 tons of space rock every day. Most of that is in the form of dust and most of that hits over the oceans (due to them covering the majority of the Earth's surface). The good news is, the larger the space rock the less frequently they tend to hit us. It takes a rock several tens of meters across to do any significant damage, and those only hit us about once every few hundred years. "Armageddon" rocks are once in 100 million year events.
Another common flash in the night has origins much closer to home, namely satellites. Watch the night sky for even 10 minutes and you're likely to observe at least one satellite. Satellites look like stars that are steadily moving across the sky. You can tell satellites apart from airplanes because airplanes blink and satellites don't. A few satellites are really spectacular. A group of satellites called Iridium communication satellites were put into orbit in the 90's to provide satellite phone service. The antennas of the satellites are particularly reflective, and when hit with sunlight just right, will become very bright (brighter than any star in the sky). This is called an iridium flare and can be identified by the way the satellite starts off very dim, then becomes very bright for a few seconds before dimming again. Another very spectacular satellite is the International Space Station (ISS). Due to it's size, the ISS is incredibly bright - unlike an iridium flare, it will maintain its brightness the entire time it is visible. You can find out when the ISS will be visible in your location by going to http://spotthestation.nasa.gov/sightings/index.cfm#.U9RBAfldXRM.
Movies often incorrectly portray comets as flying across the sky much like fireballs. Comets are actually much more subtle - they slowly grow brighter over the course of weeks and they slowly fade away, all the while moving across the sky at a pace that can only be observed by watching the comet for many days. So comets really can't be described as something that "goes flash in the night"; however, they are directly related to another "flashing" phenomenon - meteor showers.
A meteor shower results when comet leaves behind a trail of dusty debris that happens to lie in Earth's path as it orbits the Sun. When the Earth flies through the debris, the result is lots of shooting stars over a period of one or more nights - a meteor shower. There are several meteor showers that happen at the same time each year - The Perseids are a regular August meteor shower that peaks the night of August 12 and is often one of the best showers of the year. Unfortunately, this year that coincides with the full moon, making the sky much brighter and the meteors harder to see. The Orionids are a good October meteor shower and the Leonids are another of the best showers of the year and peak in November. Both of these will have little-to-no moonlight interfering. The best time to observe a meteor shower is between midnight and dawn, and the best approach is to find a dark location with a good view of the sky and to lay back and observe as much of the sky as you can - meteors will happen all across the sky. No instruments needed! Actually, telescopes will hinder your ability to observe shooting stars because they only allow you to see a tiny piece of the sky.
But remember, it doesn't have to be a meteor shower to observe shooting stars and other flashy phenomena! Get outside and start looking up!