That's the camera's shutter which is hidden at the back of the reflex mirror. It covers the digital sensor.
The shutter actuation count is the number of times that shutter has already been opened to expose the digital sensor behind it either when taking a picture or for cleaning purposes. Most factories rate their dSLR's to around 100,000 shutter actuations. This is not a definite number. The camera does not necessarily die when reaching that number. Some cameras function properly even beyond that number. A lot won't even reach it. When the shutter dies at any count, you can have it replaced and the count will be reset.
Shutter actuation count is a good measure of use of a dSLR. This is a good criteria when buying secondhand dSLR's. The closer that count gets to 100,000, the more used and abused the camera is.
As I see it, more than 50,000 is very old and not worth having unless the camera still functions perfectly and the price is below $200. A count of 20,000 is all right and expect that camera to have been used by a pro for quite a short time only so the abuse factor is there. From about 10,000 to 5,000 is just right for a camera not lower than $250. It has been used lightly and the previous owner was probably just a casual shooter. The best are those with less then 5,000 shutter actuations. Consider them almost new. Most of these hardly left the bag or cabinet and are extreme good physical condition. It's up to you to assess the camera personally though and not depend on shutter actuation count alone. This is just a guide.
So how do you find out the shutter actuation count? Take a picture with the camera. Copy the picture to a computer. Load the picture in a graphics program or an internet browser. Check the EXIF data. You can use EXIFtool, GexifView and Opanda to find that out.
Now you know!