Can pockets of air exist underwater? If yes, why don't they fill up with water, and can you breathe the air there? Like, it's not exactly atmosphere there, but an underwater cave with higher ceiling.
P.S. Possible that it has a different name than air pocket, but due to my limited knowledge of English, I have no idea.
Thanks in advance!
 A: Yes, it can exist, and the air will be pressurized to the level of the pressure that the water column exerts, which is roughly 1 bar for 10 m of depth (slightly more in sea water, because of its higher density).
There even is a practical application for this, which is called a trompe. There's an article on Wikipedia here. The gist is that air is sucked down a shaft by falling water, separates from the water stream in a chamber down below, which is, effectively, an underwater cave, and is now under pressure so when you run a pipe to the top of the chamber and put a valve on it, you have a supply of pressurized air.
As for breathing the air, at some point, when the pressure is high enough, the different partial pressures of nitrogen and oxygen will mean that breathing our normal atmosphere will deposit too much nitrogen in your blood stream. This is a problem divers face, and the reason why they don't use ordinary air for breathing beyond a particular depth. (Actually, the effects are a bit more complex, and even the oxygen itself is not harmless at higher pressures. For further information on that, see the Wikipedia article on "nitrogen narcosis").
A: A man just survived 3 days in a boat underwater in a air pocket, read this: 
http://www.slate.com/articles/health_and_science/science/2013/06/harrison_okene_s_shipwreck_air_bubble_how_could_he_survive_underwater_for.html
Okene’s salvation—the air bubble—was trapped because the overturned boat acted as a sort of diving bell, the cup-shaped chambers that have transported explorers and workers into the depths for centuries. In the fourth century B.C., Aristotle described the contraptions as enabling “the divers to respire equally well by letting down a cauldron, for this does not fill with water, but retains the air, for it is forced straight down into the water.” Years later, diving bells called caissons helped 19th-century workers construct the Brooklyn Bridge (though many died in the process).
Whether in a bell or boat, trapped air rises to the top of a concave chamber. The only way it can escape is by diffusing through the water itself, one molecule at a time. Eventually this would happen, but Okene would have succumbed to thirst, hypothermia, or asphyxiation long before his air bubble diffused into the ocean.
A: Yes, it is possible to have pockets of air underwater as long as there is something there to contain the pocket.
You can easily demonstrate this by turning a cup upside-down and submerging it in water.  If you put a napkin in the bottom of the cup before you do this, the napkin stays dry.
If you do the above experiment, but dive down in a swimming pool with the cup, you will notice that the air pocket decreases in size as you dive, and returns to normal size as you surface.  This indicates that no air is being lost.  Instead, the air is being compressed.
These observations indicate that the air pocket does fill part way up with water, but does not disappear.  This is because as the air is submerged, the water pressure on it increases.  This shrinks the air pocket's volume until the pressure in the air is the same as the pressure in the water surrounding the air.
Yes, you can breathe air in an air pocket - it's normal air.  However, you will exhaust the oxygen supply quickly if  the air pocket is small.
