The focal length defines the magnification, nothing else.
The field of view, also known as the angle of view (or more simply what you get in the picture) is defined by two things, a) the magnification - how large the objects in the projected image appear, ie focal length, and b) the size of the sensor, which determines how much of that projected image it is able to capture.
A small sensor only captures a small area of the projected image, over a narrow angle. A larger sensor captures more, over a broader angle, recording a wider field of view.
To take an extreme example, a 300mm lens is normally regarded as a long telephoto lens because on a camera with a small sensor you don't get very much in the picture. But on a large format camera that takes 10 x 8inch sheet film, 300mm is a 'standard' lens and will capture such a wide field of view it is equivalent to a 30mm lens on a crop format DSLR. Of course the two lenses are very different animals but if you were to fit the 300mm lens intended for a 10 x 8in camera onto a DSLR what you would see in the viewfinder would be exactly the same as with any other 300mm lens.
For clarity, the 35mm format (also known as full frame) is named after the width of the film, including the sprocket holes. The actual image area is 24 x 36mm. In this context, '35mm' has nothing to do with lenses.
The crop factor was originaly intended to help film users wanting to use their old lenses on new digital cameras to get an idea of what they new framing would look like. Canon use a 1.6x (linear) crop factor because they use a sensor that is 22.3 x 14.9mm which has an area less than half the size of full frame. Nikon/Sony are fractionally larger with a 1.5x crop factor and the 4/3rds format (Olympus/Panasonic) is quite a bit smaller, being one quarter the area of full frame and so it has a 2x crop factor.
To establish how much you will actually get in the picture from a given lens when used on a camera with a different sized sensor to full frame, you multiply/divide the focal length by the crop factor. The focal doesn't change, but the area of the captured image does, so that gives you the new
equivalent focal length.
For example, if you have a 50mm lens on a full frame camera, and then fit it to a 1.6x crop format DSLR, the smaller sensor will capture a smaller area of the scene, equivalent to using an 80mm on full frame (50 x 1.6 = 80).