In the early days of Pokémon, would-be Pokémon masters had very few tools at their disposal to actually catch the dang things. You could use the basic model, the Poké Ball, which worked okay. You could use Great Balls and Ultra Balls, which worked slightly better. You could also use a Master Ball, but you only got one of those, unless you used the MissingNo Cheat, which could give you more Master Balls but also a few more glitches.
Times have changed. Now, there are all kinds of different Poké Balls, all designed to suit your unique needs. Want to have an easier time catching Water and Bug types? Try the Net Ball. Want to make the Eevee you just beat the crap out of hate you a little less? Use the Friend Ball. Are you a wealthy socialite who wants a blinged out Poké Ball that works exactly the same as a regular Poké Ball but costs 10x more? There's a Diamond Ball with your name on it.
Amassed here is every single Poké Ball ever created, one for each and every type of Poké Baller.
The Master Ball is the best Poké Ball out there, with a 100% success rate. You'll only get one per game, though, so don't waste it.
Ultra Balls are excellent Poké Balls with higher catch rates than Great Balls.
The comfort level inside the Luxury Ball will make caught Pokémon much friendlier than they otherwise would be.
Quick Balls have the highest success rate when used at the beginning of a wild encounter.