LEft hand - changes which chainwheel (the big cogs at the front) the chain is on.
Right hand - changes which cog of the cassette (the cluster in the back wheel) the chain is on.
The bigger the front one you are using, the higher the gear
The smaller the back one you are using, the higher the gear.
So bottom gear is smallest ring at the front, largest at the back
Top gear is biggest ring at the front, smallest at the back
If you drive you will be familiar with high gear being useful for going fast, and low gears for going slow.
You should avoid 'crossing' the chain if at all possible. This means you should not use the gears that make the chain go too far off straight - for example don't use the largest rear cog and the largest front.
Not sure if the technicalities will help but imagine (for ease of maths) that you have a weird bike with two chainrings at the front, but only two cogs at the back.
Front chainring 1 - 48 teeth
Front chainring 2 - 24 teeth
Back Cog A - 24 teeth
Back Cog B - 12 teeth
Because the teeth are spaced evenly (if they weren't the chain wouldn't fit) we can see that:
With the chain on 1(48 teeth) and A(24 teeth) each turn of 1 means A turns around twice - so each time you turn the pedals, the wheel will go around twice.
With the chain on 1 and B (48 and 12) each turn of the pedals will make the wheel turn 4 times - harder to press the pedals, but if you can do it as fast as with 1 and A, you will go twice as fast.
With the chain on 2 and A you will get one wheel rotation per pedal stroke - easy to pedal but you will need to pedal very fast indeed to get any speed. On the other hand, it would be good for going up hill, or into a strong wind, or if you were tired, or had lots of luggage, or whatever.
With the chain on 2 and B (24 and 12) you get 2 revolutions per pedal stroke - exactly the same as if you were on 1 and A (48/24 is the same as 24/12) You will find with real bikes some gears are pretty much the same too incidentally.
The gear you use depends on lots of things - how fast you want to go, how hilly it is, how windy it is, how fit you are ... but hopefuly some of this makes sone sense and you will have a bit more of an idea of what is happening when you move the levers.