It's measured in seconds (or fractions of a second) and is often associated with the movement of your subject. For example, if you want to take an action filled shot of your cat jumping into the air you would want your shutter to move super fast - say 1/1000 of a second.
(from Sharon Montrose)
If you want to see the stars streak accross a night sky, you would want that shutter open for quite a while - say 9000 seconds (150 min!).
(from DIY Photography)
But, shutter speed also has a lot to do with exposure - the longer you leave your shutter open, the more light will hit the sensors on your camera, brightening up your picture.
Take these two from Candy Apple Smiles:
1/1000 1/15 .
Not only is the water more smooth in the 1/15 shot, but the light is significantly brighter and she even caught a sun flare - pretty :)
There's always a catch though. When you leave your shutter open for longer than about 1/60 you'll need a tripod or surface to rest your camera on or you'll get camera shake. Like this...
(from photo tuts+ - I love this guy's face "tsk tsk, you're not using a tripod")
So now we have three camera-based tools for getting better lighting in indoor photos: ISO, White Balance, and Shutter Speed. One more to go and it's a really fun one... tune in tomorrow!!