When choosing a new lens, or considering a camera with one built into it, the main consideration we have is its focal length. How much we can fit into a image, and how far we can zoom into distant details are determined by this, and together with its aperture it determines what types of photography are available to us.
How are lenses classified?
Lenses are sorted into categories depending on their focal length in relation to the format with which the are used. A 50mm lens on a full-frame body is confortable classed as 'standard', though on an APS-C body it strays into a telephoto territory.Meanwhile, on a large format camera it will fall under the term wideangle. Strictly speaking, a 'standard' lens for a particular format is one whose focal length is roughly equivalent to the diagonal of the imaging element.
For both a standard frame of 35mm photographic film and a full-frame sensor this is measured at just over 43mm - therefore anything around this length may be called 'standard'. Anything much shorter than this is classed as wide, and longer than this is known as telehphoto.
So what is focal length?
So what exactly is focal length? It is the distance between the rear nodal point of a lens and the point at which the image is focused, when the lens is focused at infinity. If you hold up a lens against a wall or your hand you should be able to see the principle in action (it's easier to do this with a telephoto lens). At one particular distance - its focal length - the image will be rendered sharply, but at others it will not.
By extending this distance, focal lengths increases; this is the principle on which a teleconverter works. But even with standard lenses, by using different optical constructions it is possible to create a lens with a longer focal than this rule would ordinarily allow. Indeed, many modern lenses take advantage of such designs for the sake of practicality.
Why can focal length change when the same lens is used on two different bodies?
So why can focal length change when the same lens is used on two different bodies? It depends not on the lens as nothing about it changes, but on the size of the sensor behind it. A smaller sensor uses a smaller portion of the lens' circle of illumination, which is the circular projection of light exiting the optic. By doing so, the peripheral areas which would ordinarily be used are cut away, leaving just the central portion.
As the size of the sensor becomes smaller, so does this central area - which is why a Four Thirds sensor has a higher focal length magnification than an APS-C one. To work out the effective focal length of a lens, the crop factor of the sensor (typically around 1.5-1.6x for an APS-C sensor) is multiplied by the focal length of the lens. So, a 50mm lens would give an effective focal length of 75-80mm, depending on the body on which it is used.
As focal length increases, the angle of view of that particular lens narrows. As an example, a 24-70mm lens will have a diagonal angle of view of 84 degrees at its widest end and 34 degrees at the other extreme, when used on a full-frame body. The same lens on an APS-C body will change these to 61 degrees and 22 degrees respectively, meaning it will reach a little further but with less wideangle coverage at the 24mm end. As such, cameras with sensors smaller than full-frame tend to bring more benefit to photographers who shoot portraits, sports and nature more than to those shooting architecture and landscapes.
What focual length do you need?
Deciding which focal length you need will largely be dictated by the subjects you photograph. One of the more popular choices in the past few years has been the superzoom lenses, so called as it can typically offer a focal range from a 28mm wideangle to a 300mm telephoto. However, to maintain image quality at all focal lengths, some form of image stabilisation system is usually required, particularly on DSLR with sensors smaller than full frame. Also, the maximum aperture will typically drop at the telephoto end, which makes it less flexible in low light. Lenses with a shorter focal length range tend to be of a higher standard - you will, for example, tend to see the professional photographer with a few high-quality lenses rather than one to cover all eventualities, even if it is more of a physical burden - but, of course, these are priced accordingly. As is the case with many other things, with lenses you do tend to get what you pay for.