What are all the different effects of the special Pokeballs?

As I understand it, catching certain pokemon with certain pokeballs provides additional effects other than an easier catch. It even shows what ball you caught a pokemon with when checking its status in your inventory or PC box, implying that it matters. (Or else why keep note of such a thing?) Which pokeballs give additional effects beyond just increasing the chances of catch and what are they?

A few Poké Balls have effects on the Pokémon in them after you catch them. These are the effects:

• Friend Ball: Increases the Pokémon's friendship to 200 when you catch it.
• Luxury Ball: Increases the rate at which the Pokémon's friendship value grows.
• Heal Ball: Fully restores the Pokémon's HP, PP, and status when you catch it.

This information comes from the bulbapedia Poké Ball page Ben Brocka linked to.