This is kind of hard to explain without being able to show you, but here goes:
(Note that this is assuming the Pokémon is in the middle of the screen.)
Drag the Pokéball off to a lower corner of your phone. While holding down on the ball, rub your finger in a tight circle until sparks start coming off. Then throw it at the Pokémon’s feet or shadow. For a ‘Nice’ curveball you need to hit inside the colored circle as usual.
Curveballs give a bit of a catch boost and a tiny bit of extra XP when you use them to catch a Pokémon. They take practice, but once you’ve got them down there is zero reason not to use them all the time.