Waves may be a purely mathematical construct and as such confusing, worrisome, and/or boring to most. Yet that doesn't mean that they don't show up everywhere.
Sound waves, light waves, ocean waves, radio waves, and "the wave" are just a few examples that we experience on a regular basis. Some more subtle examples are the vibrating waves on a string or a drum head (see oil slicks and music for more background).
Waves can be classified in a number of ways, but for now we'll just stick to two main categories: standing waves and traveling waves.
For a standing wave, think of a piano string vibrating up and down in any of the following fashions:
The alternative is a traveling wave which moves and does not have fixed points or nodes of the wave. An example of such is shown here
An example of such is when you whip the vacuum cleaner cord to get it unstuck from something. You can briefly see a short traveling wave in the cord.
Ocean waves are a form of traveling wave. Keep in mind though, that even as the wave moves across the ocean, the water itself is not moving horizontally, instead it is just moving up and down. In this sense it should start to become clear that when a wave is moving, it is typically not carrying actual stuff, but rather is carrying energy.
In the same way, as sound saves travel through air, the air particles themselves are not traveling any great distance, instead, they merely travel far enough to let the other air particles near them know how the wave is moving. So again, a sound wave is really a transfer of energy.
Finally, we get to light, which is the most confusing wave of all. Light is certainly the transfer of energy [as anyone who has ever tried to cook anything with a 60 Watt light bulb [think easy-bake ovens] knows] that propagates forward not unlike a sound wave.
That's waves.
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