When you try to catch a Pokémon, the goal is to toss your ball inside the shrinking ring around them and have it land on the Pokémon’s head or nose. But there are special catch bonuses that can help improve (though not guarantee) your chances of keeping that Pokémon locked inside that Poké Ball rather than breaking free.
Whatever bonus you’re going for, remember to tap and hold on the Poké Ball to start the ring cycle so that you can wait to toss your ball until the ring is at the correct point. (Even if the Pokémon attempts to rear back or do some sort of movement to break the circle, keep holding your Poké Ball — they’ll eventually return, and you won’t lose your ball if you keep holding onto it.)
To get the Nice 10XP bonus, you want to catch the Pokémon at the moment the circle is at its widest. That usually means throwing while the circle is at its smallest, and attempting to land straight on the Pokémon’s head.
To get the Great 50XP bonus, you want to catch the Pokémon when the circle is exactly halfway: I’ve only gotten this bonus once or twice, because it requires crazy timing. Ideally, you want to toss the ball when the circle is two sizes away from its biggest size, and land square in the center of the circle.
To get the Excellent 100XP bonus, you need to catch the Pokémon in the direct middle of the circle when at its smallest. This move is easier to attempt on smaller Pokémon (like Eevee) instead of bigger creatures, because of where their heads are positioned relative to the circle. To score the bonus, wait until the circle is pretty small, then toss your ball straight on.
To get the Curveball 10XP bonus — which also comes with a near-guaranteed success rate at catching the Pokémon — you can spin the Poké Ball in a clockwise or counter-clockwise direction until it starts vibrating, then toss it toward the Pokémon. I find this method pretty difficult to achieve regularly, so I often do an alternate method: Instead of curving, then tossing, I put the side of my thumb on the screen and throw — the contact patch makes the game think I’m spinning the ball before throwing.