It's a shame that the chart that Pengun linked shows all of those different sensor sizes in the corner. You need to think of them as centered in the middle
of the box.
A 50 mm lens is still a 50 mm lens. regardless of which camera it is mounted on. However, the camera with the APS-C sensor captures only the middle part of the image that a camera with a full-frame 35mm sensor would. The field of view is cropped
to the same field of view as a lens with 1.5 times as much focal length. That's the "crop factor".
Smaller sensors are much easier to manufacture without defects, so they're less expensive. You also have more depth of field (as if you were one stop down with the longer lens on the full-frame sensor). The other beneficial effect is that many lenses perform significantly better at the center of the image than they do at the corners. APS-C captures only the sweet spot in the middle of the image.
Take a look at the MTF chart for the cheapest EF lens made:http://www.usa.canon.com/cusa/consumer/ ... m_f_1_8_ii
Notice how the wide-open performance between 18.0 mm and 21.6 mm from the center gets pretty bad? Those are the corners of a full frame sensor. The corner of a Canon APS-C sensor extends to only 13.4 mm from the center, so all of that bad stuff at the corner of the full-frame image isn't captured by the camera. It's cropped out.
Another way to think of it is if you turned the APS-C size sensor into the portrait orientation, it would have essentially the same height as the full-frame sensor does in landscape orientation.