The puppet possesses three forms. Once you have hit its tail with three Light Arrows, it transforms into the next form. Puppet Ganon also sends out Keese to distract you, which are a good source of health and magic.
Its first form is that of a boar. It launches its arms and swings its tail to attack. To strike the tail's orb, the strings that suspend the puppet must first be severed with the Boomerang (except the main red one, which cannot be severed) to temporarily stun the monster. Otherwise, the puppet is too fast to strike accurately, if not entirely impossible.
Its second form is that of a tarantula. The giant spider will jump high into the air and spin rapidly. To successfully strike it, pay attention to the reflection in the water on the floor to see when the spider will stop spinning and crash land. Then, before it does, position yourself next to the orb and away from the monster's crushing legs and, wait for it to land, then shoot it with Light Arrows. A more advance method is to wait for it to stop spinning, then shoot it with the arrow before it crashes down, preventing Puppet Ganon from even reaching the ground (as it jumps back into the air after the strike). If done right, this form can be defeated without ever letting it land on the ground.
Its final form is that of a snake. The giant snake slithers all around the room, moving faster with every Light Arrow that hits the tail orb. If struck in the face with the Master Sword or Light Arrow, the snake will pause for a moment, leaving a small opening (which requires quick reflexes to take advantage of). Alternatively, just estimate the snake's movement and shoot Light Arrows, hoping for it to hit its target. However, there are a few easier strategies for beating him. The first is to hit him in the head with a bomb, estimating his movement. Bombs stun him long enough to run over to his tail and shoot it. The second is to plant some All-purpose Bait. It will go after it, and that will give you a clear shot. The third is to time its movements and smack it in the face with the Skull Hammer, which requires good timing as the Skull Hammer is a slow weapon and the snake moves very quickly. This keeps it in its spot long enough to roll behind it and easily shoot the orb.