Water plays two very different roles on Earth, one on its surface and the other in its interior. Both are of interest to geologists, and both can be described as cycles, but the things that drive them are very different.
The Surface Water Cycle: Seawater, Freshwater, Groundwater
The classic water cycle we all learn about in school describes the water we deal with in our lives. The ocean holds most of this water, and a fraction of it circulates through the air and the ground. You could say that seawater is the default state of Earth's surface water.
The sun and wind evaporates water from its liquid state. The amount of water vapor that can be in the air is controlled by temperature (and by pressure, but this doesn't matter outside the laboratory). Because water vapor (H2O) is a lighter gas than air (a mixture of N2 and O2), humid air has a slight tendency to rise. That in turn makes it expand and cool. When air cools enough, the water vapor in it is forced to condense into liquid droplets or ice crystals, and thus we have clouds, rain, snow and so on.
Water that rains or snows back onto the ocean closes the water cycle in a trivial sense. What we usually think of as the water cycle happens when the wind carries water vapor over the land, and the water that falls on the land makes its way back to the sea. Some of it runs quickly back in rivers and lakes, spends a few centuries in glaciers or stays a few millennia in ice caps—all of that is freshwater.
The rest of this water trickles into the ground and becomes groundwater, where it slows down dramatically. Groundwater can be millions of years old. Indeed, in places there are pockets of water that appear to have been there for more than a billion years. Nevertheless, we haven't found any that's as old as the Earth itself, so it's fair to say that groundwater, like freshwater, always finishes the water cycle.
There is a fraction of water that gets chemically changed, turning into minerals. Some are hydrous minerals, in which actual water molecules take part in the crystal structure. Gypsum is one example. Others consume water by dismantling it and keeping hydroxyl groups (–OH) in their crystal structure. Surface minerals like the clay minerals and goethite are prime examples. These become part of the second water cycle.
The Deep Water Cycle: Water of Hydration and Magmatic Water
The engine of plate tectonics takes material out of and into the Earth's deep interior in a great geological cycle. The water that takes part in plate tectonics forms another cycle that moves water into and out of the Earth's mantle, locked inside minerals.
Here I need to pause and explain that "water" is actually shorthand among petrologists for something else: the hydroxyl group. Oxygen is everywhere in the mantle and crust, but to make water the crucial element you need is hydrogen (indeed, "hydrogen" means "water-maker" in scientific Greek). The hydroxyl group is where hydrogen exclusively hides—think of it as dehydrated water—and what everyone calls the deep water cycle is really a deep hydroxyl cycle. Specialists prefer to call it a hydration cycle rather than a water cycle. Keep that in mind as I continue to use the word "water."
When tectonic plates enter the mantle during subduction, they carry with them the clay minerals and hydrous minerals that settled upon the seafloor. As the plates sink deeper, the pressure and rising heat turn these hydrous minerals back into the primary minerals that erupt from the mantle in magma. The result is that they release water of hydration.
But more important than these is the rock of the seafloor itself. As seawater penetrates into the rocks of the oceanic crust, it starts to react with the abundant mineral olivine and turn it into hydrated serpentine minerals. The rocks of this serpentinized crust release their water of hydration as the subducting plate goes down.
The point is that as we enter the mantle, there is plenty of water available for any mineral capable of holding it. And laboratory experiments have shown that most of the major minerals in the mantle—garnets and pyroxenes and olivine—can accept some water, even though their chemical formulas have no hydrogen at all. Furthermore, some of the hydrous minerals like micas are stable at far greater depths than we once thought possible.
We're still figuring out what this means in detail. But it's clear that there is as much water under the plates as there is in the oceans and glaciers and groundwater. The moisture goes down at least to the mantle's transition zone, more than 400 kilometers deep.
Water is an enlivening juice wherever it's found. In the deep Earth, water helps rock deform under stress, and it helps rock melt into magma. In magma, the "water" in mantle minerals becomes free to reconstitute itself as genuine water vapor. As magma rises and erupts onto the Earth's surface, this water is the main engine of eruptions (see "About Volcanism"). And that is how the deep water cycle completes itself.