I've had 4 PMs about this lately and still haven't found a good site to explain different lens and how they work etc.
The bigger numbers usually stated in mm is the focal length of the lens. Forget about 3x zoom or whatever you see on compact cameras, that is completely irrelevant now. It can be slightly confusing as 'normal' DSLRs will automatically record roughly 1.5 times more than the actual stated mm on a lens (i.e. a 10mm lens would actually be 15mm), but to be honest that's not worth worrying about.
10mm on a DSLR would be very very wide, wide enough to accidently get your feet in the photo, or to see more than you would see within the main vision of your own eyes...
10mm
50mm is there or thereabouts the focal length equivalent to what we see with our own eyes
50mm
100mm onwards is getting into telephoto territory, most go up to around 300mm before you start reaching silly money.
300mm
There are two main types of lens when it comes to DSLRs,
prime and
zoom. A prime lens will be locked at a certain focal length, e.g. 50mm, a zoom lens will cover a wider range, e.g. 55-200mm, and obviously you can use every mm right from the 55mm through to 200mm. To the average person a prime lens would seem a bit pointless, but the benefit is that they are usually of much higher quality and can be much 'faster' (i.e. f2.8).
Now the bit that confuses most, the 'f4.5-5.6' bit. The f number is the aperture value that a lens is capable of, essentially the lower the number the better the lens is. A lens that has a value of f1.4, f1.8 or f2.8 is known as a fast lens, a lens with values creeping up around f5.6 would be a slow lens. Many zoom lens have a range (i.e. f4-5.6) as they not capable of maintaining a fast speed throughout the whole zoom range. So a 70-300mm f4-5.6 would be f4 at 70mm, but gradually increase to f5.6 at 300mm. However an 80-200 f2.8 would be able to work at f2.8 from 80mm all the way through to 200mm.
Yes you are right to say you can shoot at faster speeds with a lower f number, 1/2000 at f5.6 may give you a very dark image in many conditions, but 1/2000 at f2.8 may be perfectly useable. Lenses get more expensive the faster they are because to maintain a fast speed requires lots of glass. For example my Nikon 80-200 f2.8 weighs over 1.5kg, where as a Sigma 70-300 f4-5.6 weighs about 500g, even though the Sigma actually zooms 100mm further than the Nikon lens.
So a benefit of a fast lens is obvious, you can use it in much darker conditions and still get useable shutter speeds. The other benefit is that a low f number (fast lens) gives you nice blurry backgrounds if needed.
300mm @ f4