It is the third boss of The Legend of Zelda: Majora's Mask, taking guard over the Great Bay Temple. By defeating Gyorg, the third giant is released, and the waters of the Great Bay are returned to normal. With its defeat, you gain its remains and a heart container. Although Gyorg does not appear as a boss character in any other Zelda game, the sharks that plague Link in The Legend of Zelda: The Wind Waker are called Gyorgs, and their appearance is a reference to how the original Gyorg supposedly wore a mask.
If you run into Gyorg while he is swimming underwater, it will take you into its mouth and bite you. The attack deals an impressive two hearts damage, so stay clear of it. Just don't get into the water until it is stunned, and you should be fine.
Gyorg will back up and charge at the platform you stand on. If it makes contact, Link will fly through the air, and possibly into the water (the attack does no actual damage, however). This is the attack that renders Gyorg most vulnerable. The long charge time allows you to take aim more effectively. Being on the center of the platform makes this attack useless, as you will not fly off the stage from the center.
Gyorg will leap out of the water and across the platform, taking anything in its path with it. If you are standing in the middle of the platform, you are likely to be caught up in this attack. If you stay on the edges, this attack won't usually affect you.
When the battle is close to the end, Gyorg will spew out some smaller fish to do some of his dirty work. Using Zora Link's barrier shield, these little fish don't have a chance.
There are a few ways to defeat Gyorg. All involve the use of magic, so make sure you have an ample supply.
Zora Link Only-
Stand near the edge of the platform. Z-target Gyorg and fire off your fin boomerangs to stun Gyorg. Once it is stunned, jump into the water and dive down. When you are close to Gyorg, use the electric barrier (R button) and then quickly retreat back to the platform. Rinse and repeat. When it releases baby piranhas, use the barrier on them too.
Link & Zora Link-
As Link, Z-target Gyorg and shoot it with an arrow to stun. Switch to Zora Link, and use the electric barrier on it. Quickly retreat back to the platform. Rinse and repeat, keeping the small fish in mind. The advantage of this method is that arrows fire more quickly, but switching masks can get repetitive.
To beat Gyorg quickly, stun it from the platform first. Then dive in as Zora Link, and use the barrier sheild. Instead of retreating, keep swimming into gyorg with the barrier on. This will repeatedly stun and damage Gyorg. After it releases its fish minions, leap out of the water. Stun it again, and go back in for the kill. The drawback of this method is that while repeatedly schocking it with the barrier, Gyorg may be able to grab you with its mouth.