How can you tell which purple ghost is real in the Forest Temple?

In the Forest Temple, after you've killed the first 3 ghosts, the fourth (purple) ghost appears near the elevator down to the boss level. It creates 3 illusions of itself and they all circle around you. If you shoot the wrong ghost, it just disappears, but if you shoot the right one, it takes damage.

I've seen videos where people hit the correct ghost every time. How can you tell which is the correct ghost?

The real Meg (the final Poe Sister), will make a quick spin as soon as she splits herself into four. Her three copies won't perform the spin, so the ghost which spins is the real meg. You can see this in action here:

Meg performs her spin shortly after the 9 second mark (she's the ghost on the left). It's bit easier to see this when she spins again just after the 47 second mark.

It is always the ghost that begins moving first.

Specifically, they will circle you at the same speed. You'll need to look for when they start spinning in place or taking some other sort of action. The first one to start the action is the real ghost.