Super Safari Ball: 1.5x catch rate in Safari Zone, only catch is you start off with 10 of these in a Safari Game at an inflated price (probably 1200 Poké)
Hyper Safari Ball: 2.0x catch rate in Safari Zone, only catch is you start off with 5 of these in a Safari Game at an inflated price than normal: (probably 3200 Poké)
For those of you wondering, admission to the Safari Zone costs only 500 Poké
Dawn Ball: Alternative to Dusk Ball, 2.5x catch rate in the daytime, costs 800 Poké and only available in the 8th, 10th, 11th towns and the Pokémon League in addition to the resident Mall.
Affection Ball: 1x catch rate, costs 1000 Poké, and doubles all affection-increasing payouts (specifically for Pokémon-Amie), available at 5th and 7th towns, the Pokémon League and the resident Mall.
Mystery Ball: Who knows what it does? 0.5x catch rate and only available from a certain NPC for 500 Poké.
Contest Ball: Can be customized (with seals, accessories, etc.) for use in the Contest Hall only. Unavailable in the Bag or anywhere else, and Pokémon in the party are automatically switched into these upon Contest Registration.
Clear Ball: A purely aesthetic ball. 1.0x capture rate and obtainable through an unknown method. Exactly what the name implies.
Starter Ball: Similar in function to the Cherish Ball, this ball holds the Starter given to the player at the beginning of their journey. Unobtainable anywhere else in the game, and also a marker that can distinguish legit Pokémon as it can't be accessed or applied to a Pokémon even through hacking attempts.