For a number of years, lenses were designed to work with 35mm (film) cameras. This means that they were designed to focus the image onto an area the size of the 35mm (slide or film) material. This area is 36mm wide and 24mm tall. Digital camera sensors come in a number of different sizes, up to and beyond this same 36 x 24mm area. For our discussion we'll only be dealing with Digital SLR's and will not include the Digital camera backs.
When light passes through a lens, it creates a round image as it reaches the back of the camera. This image is upside down and reversed from left to right. The camera's penta-mirror or penta-prism (or electronics) flips and reverses the image so you can see things normally through the viewfinder. The DSLR sensor, as with 35mm film, is a rectangle. The 35mm film corners just reach the circumference of this round image. The DSLR sensor is usually a smaller rectangle. Whatever is outside each rectangle is not included in the image captured. It's only in the most recent few years that manufacturers have been building smaller lenses designed for the smaller DSLR sensors. In the image below, the red circle represents the outer edge of the circle produced at the focal plane by a standard 35mm lens. The green circle represents the outer edge of the circle produced at the focal plane by a lens designed for an APS-C sized sensor.
Several things should be pretty obvious. You should now be able to see why you can't use a lens designed for the smaller digital sensor on a 35mm camera. The image circle for this type of lens (green circle above) doesn't cover all of the 35mm negative. If you like black corners in your pictures, feel free to use this combination (grins).
Because the APS-C sensor is smaller than the 35mm film size, only the light from the center portion of the lens designed for a 35mm camera (red circle above) reaches the sensor. This is the best part of the glass.
Light coming into the lens through the edge of the front element has to make the greatest "bend" on its path to the focal plane and actually causes the most distortions and aberrations in your picture. The smaller sensor doesn't use this light (it's outside the sensor rectangle), so lenses designed for 35mm cameras effectively act like more expensive lenses because you are only using the "sweet spot" of the lens elements, eliminating the worst parts.
When you take a 35mm camera and frame a shot to include an object from top to bottom, side to side, the light reflecting from this object fills the space of the 35mm negative or slide (36 x 24mm).
When we replace the 35mm material with a smaller digital sensor, the sensor must be placed in the same location as the original film because this is the focal plane where the lens focuses its image.
As you can see, because the digital sensor is smaller than the original 35mm negative, some of the image being focused by the lens is lost.
If you connect lines from the edges of the sensor through the center point of the lens, you get a narrower range (less total) of the object you are photographing. To get the entire object visible again, you'd need to move the camera, lens and sensor further away from the object being photographed.
If you placed a 35mm camera and a DSLR in the same location with the same lens, here's an example of what you'd see. The red rectangle represents what you'd see through the DSLR viewfinder.
Now let's print them both out as 6" x 4" photos.
Now you see how putting a lens originally designed for a 35mm camera onto a DSLR camera makes it "act" like a longer mm lens, giving you 50% greater magnification for most DSLR cameras.
So how do you know how much difference you'll see on your camera? To find the answer, you need to compare the diagonal measurement of your camera's sensor to the diagonal measurement on a 35mm negative.
We've already said that a 35mm negative is actually 36mm x 24mm for width and height. Doing a little math yields a diagonal dimension of 43.27mm. The sensor in my Nikon D80 is 23.6mm x 15.8mm for width and height. Doing the math yields a diagonal dimension of 28.4mm. 43.27 divided by 28.4 gives a value of 1.52, or about 50% increase. This means that a 100mm lens designed for a 35mm camera placed on my Nikon D80 body will give the same field of view as a 150mm lens on a 35mm camera body. A 70-200mm zoom lens will yield a 105-300mm lens field of view (35mm equivalent).
Change the size of the sensor and you change the amount of the effect. Canon makes a sensor that is the same size as the original 35mm film size (36 x 24mm) so any lens designed for a 35mm camera used with these cameras (EOS-1Ds Mk II, EOS 5D) will give the same result as with a 35mm film camera, field of view wise. They also make a sensor for the EOS-1D Mark II camera that is 28.7 x 19.1mm, giving you about 25% magnification effect. Most of Canon's DSLR's use a sensor that is 22.5 x 15.0mm, giving you about 60% greater magnification.