This is located in the Kiloude City which is only accessible via TMV. You can get a TMV Pass from Sycamore after defeating the Champion. This is a place where you can participate in different battle modes and earn BP (Battle Points) These points can be used to redeem wonderful TMs, consumable items and held items.
You have to keep battling until you lose. As you continue winning, you'll earn more BP. Losing once will end your streak and you'll have to start from zero again. Don't worry since the receptionist will heal your Pokemon every round and you can interrupt your challenge after battle and continue it later. Your progress and streak is saved.
Single Battle
Double Battle
Triple Battle
Rotation Battle
Multi Battle
As you continue your winning streak, you'll eventually have to face the Battle Leaders of each battle type. For the standard mode, you'll face them on your 20th streak. For Super Battles (these are upgraded modes of the standard battle types), you'll face them in your 50th streak with more powerful Pokemon. The LV50 limitation still apply to all Pokemon in these modes.
Battle Chatelaine Nita
Wigglytuff
Normal/Fairy
Item: Shell Bell
Purugly
Normal
Item: Big Root
Grumpig
Psychic
Item: Wise Glasses
Battle Chatelaine Evelyn
Primeape
Fighting
Item: Scope Lens
Persian
Normal
Item: Life Orb
Pachirisu
Electric
Item: Sitrus Berry
Lumineon
Water
Item: Expert Belt
Battle Chatelaine Dana
Dragalge
Poison/Dragon
Item: Bright Powder
Whimsicott
Grass/Fairy
Item: Sitrus Berry
Magcargo
Fire/Rock
Item: White Herb
Magneton
Electric/Steel
Item: Shuca Berry
Piloswine
Ice/Ground
Item: Persim Berry
Girafarig
Normal/Psychic
Item: Colbur Berry
Battle Chatelaine Morgan
Sawsbuck
Normal/Grass
Item: Shell Bell
Mantine
Water/Flying
Item: Wacan Berry
Klefki
Steel/Fairy
Item: Apicot Berry
Swalot
Poison
Item: Black Sludge
Battle Chatelaine Nita
Landorus
Ground/Flying
Item: Choice Scarf
Tornadus
Flying
Item: Yache Berry
Thundurus
Electric / Flying
Item: Sitrus Berry
Battle Chatelaine Evelyn
Suicune
Water
Item: Lum Berry
Entei
Fire
Item: Life Orb
Raikou
Electric
Item: Air Balloon
Latios
Dragon\Psychic
Item: Lax Incense
Battle Chatelaine Dana
Articuno
Ice/Flying
Item: Charti Berry
Zapdos
Electric/Flying
Item: Petaya Berry
Moltres
Fire/Flying
Item: White Herb
Regirock
Rock
Item: Luminous Moss
Regice
Ice
Item: Scope Lens
Registeel
Steel
Item: King's Rock
Battle Chatelaine Morgan
Cobalion
Steel/Fighting
Item: Maranga Berry
Terrakion
Rock/Fighting
Item: Sitrus Berry
Virizion
Grass/Fighting
Item: Coba Berry
Latias
Dragon/Psychic
Item: Lax Incense
Reference:
Stop attacking my family TJ!