The way spot metering works is pretty simple. Rather than measuring the light level across the whole scene, spot metering only takes a measurement from a small spot in the centre of the frame, typically an angle of only about one or two degrees from the centre-line of the lens. Since it ignores everything outside this area it is a very useful tool for coping with unusual lighting conditions, such as very strong backlighting when shooting into the sun or against a brightly lit window, or shooting light objects against a dark background such as a band on a stage.
The photo below was taken using standard multi-zone evaluative metering. As you can see the brightly lit background has caused the forground to be under-exposed.
By using spot metering the background is ignored, and the front of the car is correctly exposed. The spot meter measures the light in approximately the area of the red circle.
Spot metering is best used in conjunction with another little-used camera function, auto exposure lock (usually abbreviated AEL), and also with exposure compensation. To understand how best to use it, a basic understanding of how camera light meters calculate exposure will be helpful. I've written a tutorial on this subject which you can find here, but the most important part is the explanation of the zone exposure system.
On some more advanced cameras, exposure lock will have a separate control, often a button near the shutter release. On most compact cameras, half-pressing the shutter button takes a light reading which is held until the button is released or the picture is taken, however be aware that in many cases doing this will also hold the focus as well. On some cameras it is possible to make a change in the set-up menu so that holding the shutter button will only lock exposure.