In US residential wiring, you have 3 wires feeding your home off the transformer: two hots and neutral. The transformer is a 240V center tapped transformer. The neutral is the center tap.
Well, that's all fine and good, but your probably asking "what does that mean?" right about now. Well, since the transformer is 240V, you have 240V in between the two hots. The center tap divides the transformer winding in the middle. So between either hot and neutral you get 120V. Neutral is kind of a return wire for 120V circuits. (I say "kind of" because this is AC, current is flowing in both directons.)
Well, why is it called neutral, then? At your main service panel a fourth wire comes into play: ground. Ground is simply a heavy safety wire that goes to a 8ft long copper rod driven into the ground. This ground wire is connected to all metal surfaces you touch (panels, screws, metal junction boxes, metal cases on appliances, etc.) In the main panel (and only the main panel) neutral is bonded to ground. So, while neutral is not ground, when everything is correct it should be "neutral," i.e. it has no potential on it.
120V safety issues
You cannot assume neutral is grounded or safe to touch, however. By the very nature of wire, if there is a current in a wire there is a potential across it. This potential is usually very small, but is still there. The bigger problem is if the neutral connection becomes very resistive or open. Then the neutral will be live as it cannot "return" the energy to the transformer. (Again, not a totally correct analogy, but it gets the point across.) This is why you shouldn't just bond ground to neutral when you don't have a ground wire at a fixture.
When a hot wire goes open, fixtures simply fail to operate. When a neutral goes open, the fixture and the neutral between the fixture and the break go live. The fixture will not work, however, giving people a false sence of security. (No the light is not on, yes it can knock you on your butt.) This is why you want to keep your neutrals in good shape.
With the neutral you have two hot wires both capable of producing 120V to neutral. Because the hots have 240V across them (and not 0), one hot goes positive while the other hot goes negative.
So, let's say you take two loads with a resistance of 60 ohms. You hook the neutrals of the loads up to neutral, one load to one hot, and the other load to the other hot. How much energy flows through the neutral conductor? 0W. This is because the load is "balanced." The amount of energy flowing through load one is equal to the energy flowing through load two. Let's do a testcase for a specific point in time. Let's say hot 1 for load one is at +60V, and hot two for load two is -60V. Ignoring load two, load one has a current of 1A (60V/60ohms=1A) flowing from hot one through the load and back to the transformer through the neutral. We will call current into the transformer on the neutral positive, and current out of the transformer on the neutral negative. So our neutral has a current of 1A on it for load one. Load two has a current of 1A flowing out of the neutral, through load two, and back into the transformer on hot 2. So, by our signing above, load 2 has a current of -1A on the neutral. To calculate the actual current on the neutral, we add the currents for the two bulbs: 1A + -1A = 0A. Current is flowing out hot one, through bulb one, through bulb two, and back to the transformer on hot 2. Make sense?
Now, let's say you have the same setup as above, only load 1 is 30ohms. Well, now the neutral has to carry the extra amp of current. The loads are no longer balanced, so the neutral has to carry the difference. At our 60V test case, the current of load one is now 2A. The neutral current is 2A + -1A = 1A.
So, this means that the neutral only carries the difference in power between the two hots. This is also why your neutral doesn't need to be twice as heavy as your hots. Let's say you have 200A service. While you can have 400A of current flowing to 120V appliances all over your home, it is actually +200A to half and -200A to the rest. Your neutral carries 0A, not 400A.
Let's go back to our last example, with the 60ohm and 30ohm light bulbs. Let's say some unscroupulous DIYer used the conduit the feed is in for a neutral instead of a dedicated neutral wire. Let's say a clamp to the pipe came off and now we have no neutral connection. Now, we had +60 on hot 1 and -60 on hot 2. So our loads have 120V across them. Now, in this ideal test case, our loads appear as a single 90ohm load to the supply. This means that there is 1.334A flowing through our circuit (120V/90ohms = 1.334A). This also means that load 1 us underpowered by 2/3 of an an amp, while load two is overpowered by one third of an amp. If loads one and two were lightbulbs, bulb one would be dim while bulb two would be brilliant. As both loads have 1.334A flowing through them, load one has 40V across it, while load 2 has 80V across it. Remember, at this point they are both supposed to have 60V across them.
Why this is something to worry about
Our little set up above is how 240/120V applances work. In, say, your dryer, you would have a third load that is directly across the two hots. Load 3 would be your heater, load 1 would be your timer, and load two could be the light in the drum. If your neutral comes open you could toast your timer or bulb. Most appliance manufacturers actually avoid using both hots for 120V, if possible, for this very reason. You can never be sure, though.
Now, the other thing to keep in mind is your whole home functions as a 240V/120V appliance. Load 3 is your heavy appliances, the heaters in your dryer, oven, waterheater, etc.. Loads one and two are all your 120V appliances, light fixtures, etc. So while an open neutral doesn't cause too much of a headache on your dryer, it does cause a big headache in your home. Let's say your service neutral comes open, and you have 1500W of appliances on for load 1, and a 100W porch light on for load 2. That porch light is going to burn out very quick.
Why you should try to balance your load
Now it is impossible to garuantee that load 1 will equal load 2 without being obsessive-compulsive. What you can do is when you plan your load and circuits, try to balance them. This will keep current in the neutral low, which will prevent bad connections from burning open. Also, in the event neutral does fail, if the loads are close to balanced you may only see 90V/150V instead of 10V/230V across your 120V loads. Your appliances my be able to tolerate the former until you notice, they can't tolerate the latter.
A resistive neutral is a nasty little problem. It has ultimately the same effects as an open neutral, but is much more subtle. With a resisive neutral, there is a connection, but it is bad. When no current is flowing through the neutral, it appears OK. The more current flows through the neutral, the more potential develops across it by ohms law. This means that if your neutral has 25A flowing through it, and has 2 ohms of resistance, the neutral wire has 50V across it. This also means that your neutral bus in your panel is not at 0V (with respect to the transformer), but is at 50V favoring whichever load (1 or 2, as above) is heavier. This means that the lighter load will be overvoltaged.
Also note that resistive neutrals get worse with time. Since our example neutral has 25A at 50V on it, it is dissipating 1250W at some point in the wire. This point is the resistive connection, and will get hot as it dissipates all this power. As it gets hot, it will burn a little further open, and the circle continues.
How can I check for a resistive neutral
NOTE: This procedure involves probing your service panel or heavy outlet while live. This is dangerous. If you are not comfterable with this, call an electrician. They can check for this quickly and tell you exactly what you need to do to fix it.
The easiest way is with a AC voltmeter. Leaving everything on as you normally would. Check the voltage from one hot to neutral, then from the other hot to neutral. This can be done at your panel, at a dryer outlet, at an oven outlet, or at any other 240/120V outlet. If the voltages differ by more than a volt or two, you may have a problem. Call an electrician, as work on your main service must be done by someone licenced, and they have lots of experience with this.