1. Waves don't actually move water.
I know, right?? You see waves crashing into the beach and hear something vague about ocean currents. But guess what? They're not actually the same thing. The waves you see from shore only move the surface of the water, and most of that motion is up and down (if you follow a bit of water on the surface, it actually travels in a circle).
2. Water waves follow the same laws of physics as all other waves- like light and sound.
All waves are part of the electromagnetic spectrum, which excites particles (or sub-particles, or mega-particles, from electrons up to large chunks of the earth's crust and beyond). At each point in time, an infinite number of waves are interacting with each piece of matter on earth, including your body. The key is frequency: the frequency at which atoms are excited and emit light is much higher than the frequency at which a tsunami wave travels through the ocean or an earthquake travels through the ground.
3. Surface water waves can cross over each other without crashing.
The physical laws governing wave behavior support addition of waves (superposition)- meaning that when one wave travels toward another, their overall effect can essentially add up to one huge wave for just a moment. This is sometimes difficult to find, because waves have to be traveling from two sources facing each other. But if you've ever watched a wave crash on a beach, retreat back to the ocean, and somehow out of nowhere the next wave in line is larger than it was before, this phenomenon may be contributing. You may also see a wave traveling away from the beach- this is the effect where they pass through each other. If they are from opposing directions, often some energy is lost as they pass through each other, but often it's not enough to dissipate them completely.
4. Waves can travel on top of (surface) or within (internal) the water.
Internal waves are nearly impossible to see from above the water (or below the water for that matter). By definition, water waves travel at interface of two different fluids. The most common we see are surface waves. But eventually some scientists realized that since air is also a fluid, this principle could be extended to include interfaces within the water. For example, the water at the surface is heated by the sun during the summer. At some point, the sunlight can't penetrate (hence those silly fish with flashlights hanging off their heads to attract prey), which means the water there is much colder. There is a transition point (depth) where this occurs. As it turns out, a lot of energy travels through the ocean along this transition- so much that we can't really approximate it yet!
5. Internal waves cause a lot of mixing, which is really important for the ocean food chain.
As waves travel, they lost energy. Internal waves are particularly good at losing energy in the form of turbulence, which mixes up the water. Because they often occur along the transition between warm, sunny water and cold, dark water (which tends to have more nutrients), internal waves bring a lot of nutrients up to the plankton living nearer the surface. These plankton grow and grow because they get these nutrients, forming the base of the ocean food chain (plankton get eaten by little fish, which get eaten by bigger fish, which get eaten by bigger fish and/or marine mammals, etc).
As it turns out, the ocean is incredibly important for our food supply as humans- around the world, people consume a whole lot of fish- but the coastal ocean in general is actually one of the most biologically productive regions on the planet. That's good for plants and animals living in the ocean, and is also good for us.