f numbers represent the size of the aperture, the hole that allows light through to the sensor or film. Small f numbers mean a big hole, so they let more light through and you can use faster shutter speeds. This has the effect of giving a shallow depth of field, ie how much of the shot will be in focus, with the background and foreground blurred. Big f numbers mean small holes, so you need to keep the shutter open longer to let as much light in. More of the shot will be in focus as the depth of field is much greater.
Prfoessional glass tends to be big and expensive to get those elusive small f numbers (most of my zooms are a constant f2,8 through their range, but they are eye wateringly expensive), unless its a prime (non zoom lens) which are much less complicated and tend to be cheaper. A 50mm f1.8 is a very good starting point for a large aperture lens. Lots of light gets in, some very shallow DOF shots are possible and they tend to be about £80, bargain!
There is also a "sweet spot" where a lens performs at its best. This is usually around f8. There's also the complicating factor of chromatic aberrations at very small apertures, usually f13 and smaller.
If your camera has an aperture priority mode, then you basically choose the depth of field by choosing the f number, and the camera will do all the maths to work out your shutter speed. Hold the camera still, and change the f number and watch the effect on the shutter speed.