Effects:
> Elevated chance of critical hits.
> Elevated chance of avoiding attacks.
> Extra Experience gain.
> Shaking off status conditions. (Albeit Freeze)
> Attacks doing extra damage.
> Enduring attacks that could otherwise make it faint and surviving with 1 HP. [(Source)][1]
Exact percentages aren't known, but effects occur more often at Affection Level 5 than Affection Level 4.
Possible messages:
2 Hearts:
(Pokémon) is reassured by (Trainer)'s familiar scent.
(Pokémon) seems a bit excited.
(Pokémon) seems a little out of it...
(Pokémon) can't wait to begin.
3 Hearts:
(Pokémon) is awaiting 's directions...
(Pokémon) seems to be interested in the other Pokémon's scent...
(Pokémon) is remembering the first time it met (Trainer)...
(Pokémon) seems curious about what is going on around it.
(Pokémon) is reassured by (Trainer)'s familiar scent.
4 Hearts:
(Pokémon) is awaiting (Trainer)'s directions...
(Pokémon) is looking at (Trainer) with trusting eyes!
(Pokémon) is bursting with enthusiasm for battle!
(Pokémon) and (Trainer) are breathing in perfect sync with one another!
(Pokémon) trusts (Trainer) to come up with the best strategy!
5 Hearts:
(Pokémon) is awaiting (Trainer)'s directions...
(Pokémon) is looking at (Trainer) with trusting eyes!
(Pokémon) is bursting with enthusiasm for battle!
(Pokémon) and (Trainer) are breathing in perfect sync with one another!
(Pokémon) turned back towards (Trainer) and nodded in understanding!
(Pokémon) seems curious about (Trainer)'s bag...
(Pokémon) trusts (Trainer) to come up with the best strategy!
3 Hearts and above also says the message "Go on, (Pokémon)! I know you can do it!" when switching in, "That's it! I knew you could do it" when hitting with a super-effective move, and "Come on back, (Pokémon)! You did great out there!" when recalling it.
Source: Above Link
[1]: http://bulbapedia.bulbagarden.net/wiki/Pokemon_Amie#Affection_effects