Appearance-wise, I'd say good old Poké Ball is my most favorite. It's simplistic, elegant, and is generally the most known type of ball there is. Chances are, those who are not exactly a fan of Pokémon would still know how Poké Ball looks, as they might've seen it somewhere before.
In terms of mechanics, I'm actually pretty fond of Master. With this, you're able to capture any Pokémon with only one throw (even legendaries), which, if you ask me, is a very powerful thing. You're given this ball throughout your journey only one time in each game-play (you could still win them by drawing lottery and whatnot, but chances aren't always very likely), which I believe also helps explain its importance.
Alongside above, I also like Dusk Ball, as they make it easy to capture Pokémon in dark places such as caves (use them when encountering legendaries, since that's where they usually reside).