Pokémon X and Y Walkthrough, Pokémon Move Sets: Azumarill
(Please note that the recommendations below are largely made for in-game play. An Azumarill in a competitive environment will likely make use of different moves to accommodate for smarter, more adaptive opponents. If you have a different bred of Azumarill you'd like to share, toss it in the comments.)
For a long time, Azumarill was a fairly meh pokémon. It looked a bit funny, true, and it had the benefit of a lot of HP, but beyond that? This thing was just a damage sponge. But then the sixth generation came along, and with it Azumarill gained a significant boost in usability. It's still a teensy bit gimmicky at times, but this egg-shaped can do some surprising things if properly trained.
Type: Water / Fairy
Azumarill starts off as the pint-sized Azurill. Raise Azurill's friendship to the max and it will change into a Marill, then level the Marill to 18 and it will transform into the full-ledged Azumarill.
Bug, Dark, Fighting, Fire, Ice, Water. Immune to Dragon.
Electric, Grass, Poison.
Azumarill has received a slight statistical boost in the new generation. Its HP remains on top at a solid 100, which is bolstered by decent (though fairly average) Defense and Special Defense stats. Attack, Special Attack, and Speed are all pretty low comparatively. Not that impressive if you just look at the stats... but, fortunately, Azumarill has a lot more going for it than just base numerics.
Thick Fat: The pokémon takes half damage to fire- or ice-type moves. On other pokémon this would be very helpful; on Azumarill it just cuts down on moves that aren't terribly effective against it in the first place. Skip it.
Huge Power: The pokemon's Attack stat is boosted by half. This is a fantastic ability, and accounts for the massive jump in Azumarill's viability in the new generation alongside its induction into the fariy-type fraternity. Absolutely stick with Huge Power.
Sap Sipper: If hit by a grass-type move, the pokémon will turn the damage into HP instead. This would be a heck of a lot more useful if Azumarill was still a pure water-type pokémon. As it is, Sap Sipper is mainly good against opponents who don't / have forgotten the addition of fairy-type to Azumarill. Sap Sipper is a hidden ability.
Azumarill has become extremely popular in this generation, and with that popularity has come a move set that some folks swear by. Most commonly when you see an Azumarill it will start off by bolstering its moves with Belly Drum. That done, it will assault its victims with a combination of Waterfall and Play Rough. If something fast by fragile comes along, the breedable move Aqua Jet jumps into action to take priority. The result is a fairly brutal wall which is fairly difficult to kill if it doesn't wipe out foes in a single hit.
If you want an Azumarill that's a bit less predictable but still powerful, you can trick it out with some alternate moves instead. Keep Belly Drum for maxing out Attack, then drop in at least one other move for variety. Superpower becomes utterly devastating under Azumarill's control (though I recommend it as a departing move only), as does Giga Impact. Brick Break is a great choice for wiping out steel-types that would threaten its fairy-typing, Dig is fun for avoiding attacks, and Return is all-around good. I recommend still tricking out your Azumarill with Aqua Jet, regardless of your other options - there are just too many speedsters that can threaten it not to go for priority.
EV / Super Training
As most Azumarill builds emphasize its ridiculous Attack boost, Attack itself is a solid choice for EV training. This stat is pretty lacklustre if you don't have Huge Power. Beyond that you'll want to work on Azumarill's HP, which will make up for its equal, and equally meh, defensive capabilities.
Catching an Azumarill
Azumarill itself can be caught by fishing with the Super Rod on Route 22. You probably won't want to work with a full-fledged Azumarill, though, in which case you can snag Marills on Route 3 while Surfing. You can even get Azurills, if you like, by hunting around in the grassy areas of Route 3 and Route 22. You'll have to breed through other water-types to get Aqua Jet, though, so don't expect one of these creatures caught in the wild to fulfill your needs fully.