The numbers are the amont of teeth on the chainrings and sprockets. eg 53x13 or 42x15. The larger number typically represents the chainrings and smaller the sprockets. Some chainrings and sprockets acutally have the numbers stamped on them and so you maybe able to read them.
To put it simply, a bigger gear (higher number of teeth) in the front, means a more difficult ride, and a smaller gear (lower number of teeth) in the rear also means a more difficult ride.
Most people have bikes with a 53/39 set of chain rings and a 12-23 set of rear cogs. If you want high gears (go fast or work hard), you want to be on your big chain ring. If you want lower gears for a headwind or a hill you want to be in your small chain ring or middle ring for a triple. If you have a triple, and you are on a big honking hill, then you go to your smallest ring (if you have a double, you will just have to make do with what you have). Remember, the 53/39 example is a common one but bikes come with all sorts of chain rings like 52/42, 52/42/30 and others.
Now on the rear cluster, the bigger the cog, the lower the gear. The rear cluster will have a whole range of gears or cogs. They will generally be 12,13,14,15,17,19,21,23,25 or something like that. To get the biggest gear (hardest to peddle) you want to be on the small cog, 12. To get the lowest gear, you want to be on the 25 cog. Now it is important to realize that bikes come with all sorts of rear clusters - some with 7 cogs, some with 8 some with 9 and some even with 10 so your situation will vary. The example is just a common one.
The thing is that old bikes used to be direct drive. One pedal stroke produced one RPM. They problem was that a wheel had to be gigantic to produce a high enough speed to be of any use. That is why the old fashioned bikes had those huge front wheels. Those things had problems because "taking a header" was not a good thing.
That led to the introduction of the "safety bicycle" which is what we see today. Basically it is a bike with two equal sized wheels and gears that allow you to generate more then 1 wheel revolution per pedal revolution. You do that with a gear. Say you want to produce 4 wheel revolutions per pedal revolution. You can do this by having a gear with 48 teeth at the pedal (the chain ring) and 12 teeth back at the wheel. You connect them with a chain. That way, when you turn the crank (where the pedals are) one revolution, you force the wheel to make 4 revolutions (48/12). If you want to have 2 speeds you can put another gear back in the rear say an 18 tooth gear and this will produce 48/18 = 3 revolutions per time you turn the pedals. As you can see, you can keep adding gears and produce more "speeds".
The speed of a bike is determined by how far you go per revolution of pedalling and how many times you pedal in a minute (rpm). Most bikes have wheels that are 27" in diameter. If you have a 48/18 combination, you can multiply the 4 times 27 to give you chain inches or 108 in this case. Why use this? Well when bike makers were trying to get people to buy the new fangled "safety bikes", people were used to riding the "ordinaries" or the bikes with the huge front wheels. On an ordinary, your maximum speed was limited by the diameter in inches of your drive (front) wheel. With a safety bike, you can set up the gearing any way you want and bikes were sold on the basis of how many chain inches their gearing was because the public, having ridden ordinaries, understood the number.
Here is a somewhat mathematic guide to gearing.
You take the teeth on the chain ring like 48. You divide it by the teeth on the rear cog like 24. You get a 2. You multiply this by the diameter of your wheel (27"). This gives you a gear of 56 chain inches. OK so what good is that? Well, 100 chain inches at 100 rpm will produce a speed of 30 mph. If your rpm is more like 90, you would get 27 mph at 100 chain inches. If your rpm is 100, you would get .56 times 30 = 16.8 mph in a 56 chain inch gear. If your rpm is 90 you multiply the 16.8 times .9 = 16.0 mph. Using this math you can figure up the chain inches for all your gears and the mph you would cruise at at your typical rpm. You can also use this as a way to determine your rpm if you do not have a computer with cadence. Say you are in a 50 chain inch gear. You know that at 100 rpm, you would cruise at 15 mph. You find that you typically cruise at 14 mph. So your rpm will be = 14/15 times 100 = 93 rpm.