Imagine an earthquake (which is a wave) coming to you. Or rather, let us imagine there are two earthquakes. The one coming from the right wants to make you move up while another one from the left arrives and it wants to make you move down (at that particular moment). Can those two earthquake waves cancel?
Yes, in fact all types of waves can add up or cancel and this story about earthquakes waves cancelling is not as crazy as it sounds. In many earthquakes, you will often find buildings untouched while right next to them you find only rubble. Usually this happens when the earthquake wave bounces on something and interacts back with itself. In some lucky places, it can cancel completely.
Adding two or more waves together to get cancellation (smaller amplitude) and reinforcement (bigger amplitude) is called interference. Below is an example of interference from music.
If you play music, you may know that you can tune your instrument by playing your note against a standard note and listening for beats (regular thumps). What you are doing when you tune is adding two waves with nearly, but not exactly, the same pitch (frequency). You can see this in the figure below. Over time you will hear the two sound waves cancel and then when the two waves overlap the sound will double and you hear a louder sound. The alternation of the quiet and loud volume you hear due to this interference is called beats.