From Bulbapedia, the community-driven Pokémon encyclopedia.

A Cool Couple (Japanese: エリートカップル Elite Couple) is a pair of Pokémon trainers that first debuted in the Generation III games. They are generally depicted as a pair of male and female Cooltrainers.

They use complementary Pokémon of opposite genders in Double Battles.

In Generation IV the pair of oppositely gendered Cool Trainers is called Double Team.

Appearance

Cool Couple Sprites

Image from Pokémon FireRed and LeafGreen

Generation III-introduced trainer classes
Ruby, Sapphire and Emerald
Aqua Admin | Aqua Leader | Arena Tycoon | Aroma Lady | Battle Girl | Bug Maniac | Collector | Dome Ace | Dragon Tamer Expert | Factory Head | Hex Maniac | Interviewer | Kindler | Lady | Magma Admin | Magma Leader | Ninja Boy | Old Couple | Palace Maven | Parasol Lady | Pike Queen | Pokémon Ranger | Pokémon Breeder Female Psychic | Pyramid King | Rich Boy | Ruin Maniac | Salon Maiden | Female School Kid | Sis and Bro | Team Aqua Grunt | Team Magma Grunt | Triathlete | Tuber | Winstrate | Young Couple
FireRed and LeafGreen
Cool Couple | Crush Girl | Crush Kin | Painter | Team Rocket Grunt
Colosseum and XD
Area Leader | Athlete | Bandana Guy | Bodybuilder | Casual Dude | Casual Guy | Chaser | Cipher | Cipher Admin | Cipher Cmdr Cipher Head | Cipher Peon | Cipher R&D | Curmudgeon | Deep King | Fun Old Lady | Fun Old Man | Glasses Man | Grand Master | Guy Hunter | Kaminko Aide | Lady in Suit | Matron | Miror B.Peon | Mt.BtlMaster | Myth Trainer | Navigator | Newscaster | Pregymleader Rider | Robo Groudon | Rogue | Roller Boy | Shady Guy | Sim Trainer | Spy | St.Performer | Supertrainer | Team Snagem Snagem Head | Thug | VR Trainer | Wanderer | Worker