How Do 18 Speed Bikes Work?

For my daughter's 9th birthday my wife and I gave her a new bike as she had outgrown her old bike.  At her age and height, most bikes are speed bikes with gears that can be changed for increased speed or ease of pedaling.  We ended up choosing an 18 speed bike that has 3 front gears and 6 back gears.

How do gears work?  You switch the front and back gears by turning the left and right handle bars.  That's easy enough, but why does this change the ease of pedaling or make it harder to pedal?  If you take a look at the gears you'll notice they are different sizes.  Turning the handle bar moves the chain from a smaller to larger gear or vice versa.

If you're trying to pedal uphill, you may find that it is increasingly harder to pedal the bike, so you can downshift to a lower gear (smaller sized gear) for easier pedaling.  You have to pedal more to keep the bike moving at the same speed, but the pedaling is easier.  This is the trade-off.  It's easier to pedal, but you must pedal more.  The opposite is true if you shift to higher (larger) gear.  It's not as easy to pedal, but you don't have to pedal as much to maintain a specific speed.  

Getting your first 'speed' bike is a rite of passage in childhood and my 9 year old loves it!  Fortunately the weather has been nice and she's already been able to get out and ride it several times!