Gear changing is about moving the chain to another chain cog. There are front and rear chain cogs, and different reasons to change either.
Now there are two parts to gear changing. First, you need to know how to change a gear and then you need to learn when to change a gear.
Gears and Changing
The right-hand lever(s) operates the rear gear. Depending on your bike, there could be up to 9 gears (or cogs) you could use. The bigger the cog, the smaller (or easier) the gear - ie. the biggest cog will be 1st gear.
The left-hand lever operates the front gear, which typically has 3 gears (or cogs) to use. In complete opposition to the rear gears, the bigger the cog, the bigger (or harder) the gear - ie. the smaller cost will be 1st gear.
And you don't have to be too concerned about knowing which cog equals what gear. Most bikes through to intermediate level have numbers on the gear shifter.
Changing gears doesn't require a clutch either! Most shifting mechanisms use levers to change up and down through the gears.
This doesn't apply to singlespeeds though - you mad buggers only have one cog at the front and one at the back!
Using Your Gears
Knowing when to change and what gear to change to is one of the more important aspects of MTBing. It will usually make the difference between riding up a hill or walking the bike up!
So what's shifting about? In a nutshell, you shift down to make it easier to pedal (ie. going up a hill) and you shift up to make it harder (ie. on a flat or downhill where you're pedalling too fast).
And what's with the front and rear gears? Essentially, you should think of shifting as a tuning exercise. The front gear matches the terrain you are riding and and the rear gears allow you to finetune along the way. Basically, the front gears make the biggest change while the rear gears make smaller incremental changes.
So here's a couple of examples.
1. You're pedalling along a flat trail and you see a hill appear in front of you. It's not a steep hill, just nice and gradual. What gears?
Check what front gear (or cog) you are in. If you're riding a flat trail, most likely you'll be in the big gear. Also check what gear you are in on the rear. You shouldn't be in a small gear but take note, just in case.
Most people need to change down one cog on the front gears if they are climbing a gradual hill. This is something you'll get the feel for though. Just before you start climbing the hill, you want to move down that one gear at the front. This is the front gear that matches the terrain in front of you. As you start climbing, you may find you'll need to move down through the rear gears until you find the right gear for you. The aim on climbing is to find a gear that allows you to pedal fairly easily while not spinning ridiculously.
2. Aarrgh .. you're faced with a steep hill!
Again, check which front and rear gears you are in. If it's a steep hill, you'll usually need to move down to the smaller front cog, if you're not already there. This is called granny gear! As you start climbing, you'll probably find you'll need to move down through the rear gears until you can pedal reasonably easy. This is called spinning up a hill and is the best way to climb, unless of course you'd prefer to walk!
And Some Don't Do's
Don't:
1. Use opposite gear combinations, ie. the big front cog and the big rear cog, or the small front cog and the small rear cog. These combinations run your chain at an angle, and aren't healthy for your bike's drivetrain. What's worse, it sounds awful and makes you look like a novice!
2. Try to change down on the front cog while on a steep climb unless you absolutely, positively have to. You can end up with nasty phenomena like chainsuck, bent cog teeth, etc. Nasty!
Knowing how to change gears is a very handy skill. Practice and learn!