When you throw a PokéBall at a Pokémon to capture it in Pokémon Go, you can increase your chances of catching it depending on how you throw. The white circle around the Pokémon determines the quality of your throw, and the one inside that constantly shrinks determines the accuracy. If you throw your PokéBall inside the shrunken circle, you may receive a Nice, Great, or Excellent throw. Of the three, the excellent throw is the most difficult to achieve.
To get a great throw, you have to wait until the shrinking circle almost reaches the dead center of the larger circle. It creates a small icon in the center of the Pokémon, which makes it quite difficult to kill to death. You can only make a great throw by having your PokéBall throw a land in the small circle, not by waiting for the smaller circle to appear and then attempting to capture the Pokémon.
An early strategy for some Pokémon Go trainers was to wait until the circle got extremely small and then remove their finger from their PokéBall. Unfortunately, you can't do that anymore and it resets it.
When the icon gets small enough, you throw your PokéBall on the one in front of you and hope it reaches the icon. You'll know if he hit the smaller one, because he'll have a notification and tell you if it was a nice, great, or great throw. You will likely switch between a Large or Excellent roll, especially when trying to wait for it to get smaller.
A great way to rack up great throws is to do so on more common Pokémon, such as the ones you encounter while walking around or hanging from new PokéStops. It is much harder to do this against Legendary Pokémon and Research Task encounters. When you roll the Excellent roll, it increases the chances of catching that Pokémon.
It takes a little practice to nail a great shot. Due to the small size of the interior icon, this won't be easy, and you probably won't want to try when the target Pokémon is far away. Try to wait to meet a Pokémon that is quite close to you.