Constructive and destructive interference:

Above: The anatomy of a wave - remember it, it's really a useful thing to have in your brain for a Physics exam.

If waves are moving oscillations (I promise they are, see this post), what happens when they run into each other? 

That might seem a fairly obscure question to ask, but it turns out that:
 
A) Asking obscure questions that no-one else thinks to ask is a big part of science, and occasionally leads to amazing discoveries (or very large explosions).

B) This question is fairly important for a lot of problems*. Especially if you're a sailor caught between two tidal waves.

Physicists don't just have an explanation for two waves colliding (here's a hint: The sailor's in trouble), they have a principle (which is much classier). It's called the principle of superposition, and it goes:

The total disturbance at a point where waves meet is the sum of the disturbances due to each of the waves. 

'The disturbance' is the oscillation of the wave, so when two waves meet at a point  you can find how much the medium they are travelling in is disturbed by taking the  size of each wave at that point and adding them together (be very careful with your plus and minus signs!)

Let's say two waves on a string  run straight into each other - that's a very simple example but we'll look are more complex situations in a later post. At any point where the waves are meeting they will add together, and what you actually get depends on whether the waves are both of the same sign or not, and how big they each are at that point. 
To keep things simple lets look at just two situations: two waves where the peaks and troughs line up, and two waves where the peaks align with the troughs. In the first situation the two waves have the same sign everywhere, so are pulling the wave medium in the same direction.
If the waves are pulling the same direction and are aligned then you just add the amplitudes of the waves at every point, and the wave oscillation will be bigger at every point (which is called constructive interference), like in the picture below...

Above: When two waves meet and the peaks and troughs align we get a bigger wave - it's really just that simple.

 

.... if the waves are pulling in opposite directions at that point the total disturbance will be smaller (destructive interference).....

Above: Where a peak lines up with a trough the waves will cancel out, either partially (for different size waves) or completely (for waves the same size).

 

And, thanks to the magic of Youtube, I can show you slow-mo examples of both. As I like destroying things, lets look at destructive interference first:

Above: When the two pulses meet in  the middle they momentarily cancel out, in a process called destructive interference
 

See how, right in the middle, the two waves both seem to disappear? That's because, at that moment, the trough of one wave is right over the peak of the other, and visa versa - the result is that the two waves momentarily cancel out, and almost vanish.  But when we get two peaks (or two troughs - it's the same thing just upside down) that line up we don't get a cancelling effect reducing the height of the wave, we get the height of both waves combined:



Above: A very similar situation, where two waves run into each other - except that, now, the two pulses are pulling the medium the same way, so they reinforce each other causing a big peak in the middle.

 

The difference between the first and second videos is the direction that medium (the string in this case) is being disturbed - if the two waves are disturbing the medium in the same direction at that point the overall disturbance is bigger, and if they're disturbing it in the opposite direction the overall disturbance is smaller than either of them. This is why it's very important to keep track of your plus and minus signs, because these are what's telling you which way the wave is pulling: If you sum two waves that have an amplitude of 3 cm you can have a resultant combined disturbance of 3cm + 3cm = 6cm, when two peaks align. But you can also have a resultant disturbance of 3cm + (-3cm) = 0 cm when a peak and a trough align.

That's basic constructive and destructive interference.... for very simple waves that just go backwards and forwards on a string. As we go on we'll meet more complex problems involving waves in two, or even three dimensions, but the underlying principle of superposition won't change: 

The total disturbance at a point where waves meet is the sum of the disturbances due to each of the waves

Or, in other words, where two waves meet their amplitudes will add, and the resulting disturbance will be bigger if they both have the same sign at that spot, and smaller if they have different signs at that spot. It's important to remember that idea, and be careful with the plus and minus signs.
Why learn all this? Firstly, in the world, you can get paid for actually knowing something useful (you didn't think teachers were torturing.. er... training.. your brain in that classroom for nothing did you? ). Secondly, if this doesn't come up on your exams at some point I'll eat my hat.
And it's a nice hat.

* For example, the one's you'll get asked in your exam.