The shutter count is how many pictures the camera has already taken. Also known as actuations. This number is important because the shutter mechanism is usually the first thing to fail on a (digital) SLR - the shutter blades are delicate and wear out.
Low end cameras, like Canon Rebels and the Nikon D40, D80 are rated for 50,000 shots.
Mid-level cameras, like the Canon 30D and Nikon D200 are rated for 100,000 shots.
Professional bodies are rated for 250,000 shots.
Usually a camera lasts a lot longer, but you'd have no reason to complain if it died after it reached the number it was rated for. And since a repair will set you back around $300, the lower the shutter count is on a used camera, the better.
You can check the actuation count yourself for most camera models with a free program called Opanda IExif Viewer. Open the most recent picture from a camera with this software and it will show you a ton of 'hidden' information.
And finally, put the shutter count in perspective - even if you take 500 pictures a month, it'll take almost two years to add another 10,000 shots to the counter... by which time you're probably ready for an upgrade anyway.