Waves, when they coincide, add to each other. This is called interference.
You've probably dealt with some kind of interference before. When you're listening to a radio song, but there's a power station close by, the electromagnetic waves from the radio add to the noise from the power station to create a less-than-clear sound. The same thing happens in a choir when two people sing together, the sound waves emanating from their voices add together to produce one sound wave that can be recorded onto a CD.
There are two important kinds of interference that need to be mentioned: constructive and destructive. Constructive interference occurs at a point when two waves add together, to increase the amplitude of the wave, causing the sound to grow louder or the electromagnetic signal to grow stronger. Similarly, when two waves are opposite in phase (one is high and the other is low), they add together to create zero strength at that point. This is called destructive interference.
Most waves just add together to create interesting patterns.
The graph below adds two waves. Play with the frequencies and amplitudes below to see how interference really works!
Also see the page on diffraction to see another kind of interference.