Counterpicking is the act of choosing a character or stage in order to have the advantage on your opponent in the next round.
Character counterpicks may be made after a lost match in a tournament ruleset. For example, after losing to a Jigglypuff player in Melee, a smasher might counterpick with Fox. Fox counters Jigglypuff because his up smash and up aerial are his best killers, and Jigglypuff will die from said moves at around 55% since it is a light and floaty character.
Stage counterpicks are much the same, except that the losing player chooses the stage to be played on instead. For example, in SSBM, a player may counterpick Mute City when playing against Falco, as most of the time the stage presents unfavourable conditions for Falco's recovery. Or, a Marth player may counterpick Yoshi's Story, as it is a small stage with close-together platforms - things which work very well with Marth's playstyle (easy tippers and less projectile distance available). Stage counterpicks are often regulated by Dave's Stupid Rule.
Brawl Stage Counterpicks|Reference: http://www.smashboards.com/showthread.php?t=187501
Bowser
Best: Norfair, Battlefield, Castle Siege
Worst: Jungle Japes, Brinstar
Captain Falcon
Best: Battlefield, Pokemon Stadium 1, Norfair
Worst: Final Destination, Smashville, Jungle Japes
Charizard
Best: Halberd, Pokemon Stadium 1, Castle Siege
Worst: Battlefield, Norfair, Final Destination
Diddy Kong
Best: Final Destination, Battlefield, Smashville
Worst: Norfair, Brinstar, Rainbow Cruise
Donkey Kong
Best: Brinstar, Jungle Japes
Worst: Frigate Orpheon, Rainbow Cruise
Falco
Best: Jungle Japes, Final Destination
Worst: Norfair, Rainbow Cruise, Luigi's Mansion
Fox
Best: Smashville, Final Destination
Worst: Delfino Plaza, Lylat Cruise, Luigi's Mansion
Game & Watch
Best: Rainbow Cruise, Green Greens
Worst: Final Destination, Yoshi's Island (Brawl), Luigi's Mansion
Ganondorf
Best: Pirate Ship, Norfair, Brinstar
Worst: Rainbow Cruise, Port Town Aero Drive Ice Climbers
Best: Final Destination, Smashville, Pictochat
Worst: Rainbow Cruise
Ike
Best: Green Greens, Pirate Ship
Worst: Jungle Japes, Frigate Orpheon, Luigi's Mansion
Ivysaur
Best: Green Greens, Norfair
Worst: Rainbow Cruise, Frigate Orpheon, Port Town Aero Drive
Jigglypuff
Best: Jungle Japes, Frigate Orpheon, Smashville
Worst: Battlefield
King Dedede
Best: Green Greens, Pokemon Stadium 1, Delfino Plaza
Worst: Jungle Japes, Brinstar, Norfair
Kirby
Best: Jungle Japes, Rainbow Cruise, Pokemon Stadium 1
Worst: Pirate Ship, Yoshi's Island (Melee)
Link
Best: Rainbow Cruise, Norfair, Smashville
Worst: Frigate Orpheon, Jungle Japes
Lucario
Best: Luigi's Mansion, Jungle Japes, Frigate Orpheon
Worst: Green Greens, Rainbow Cruise
Lucas
Best: Norfair, Brinstar, Luigi's Mansion
Worst: Lylat Cruise, Yoshi's Island
Luigi
Best: Brinstar, Norfair, Frigate Orpheon
Worst: Final Destination, Jungle Japes
Mario
Best: Yoshi's Island, Smashville, Battlefield
Worst: Jungle Japes, Pirate Ship
Marth
Best: Battlefield, Delfino Plaza
Worst: Brinstar, Jungle Japes
Meta Knight
Best: Luigi's Mansion, Rainbow Cruise, Delfino Plaza
Worst: Yoshi's Island (Melee), Green Greens, Final Destination
Ness
Best: Delfino Plaza, Brinstar, Pirate Ship
Worst: Yoshi's Island, Yoshi's Island (Melee)
Olimar
Best: Luigi's Mansion, Norfair
Worst: Rainbow Cruise, Jungle Japes, Frigate Orpheon
Peach
Best: Jungle Japes, Luigi's Mansion, Brinstar
Worst: Lylat Cruise, Rainbow Cruise
Pikachu
Best: Norfair, Final Destination
Worst: Luigi's Mansion
Pit
Best: Norfair, Lylat Cruise, Final Destination
Worst: Pokemon Stadium 1
R.O.B.
Best: Frigate Orpheon, Luigi's Mansion, Jungle Japes
Worst: Distant Planet, Norfair, Brinstar
Samus
Best: Battlefield, Final Destination
Worst: Jungle Japes, Yoshi's Island
Sheik
Best: Battlefield, Halberd
Worst: Jungle Japes
Snake
Best: Norfair, Halberd, Smashville
Worst: Jungle Japes, Rainbow Cruise
Sonic
Best: Rainbow Cruise, Yoshi's Island
Worst: Brinstar
Squirtle
Best: Delfino Plaza, Battlefield, Norfair
Worst: Jungle Japes, Pokemon Stadium 2, Pictochat
Toon Link
Best: Rainbow Cruise, Pirate Ship
Worst: Castle Siege, Luigi's Mansion
Wario
Best: Rainbow Cruise, Jungle Japes, Frigate Orpheon
Worst: Final Destination
Wolf
Best: Smashville, Yoshi's Island
Worst: Lylat Cruise
Yoshi
Best: Halberd, Final Destination, Rainbow Cruise
Worst: Jungle Japes, Green Greens, Frigate Orpheon
Zelda
Best: Smashville, Battlefield, Luigi's Mansion
Worst: Lylat Cruise, Delfino Plaza, Halberd
Zero Suit Samus
Best: Pokemon Stadium 1, Lylat Cruise, Battlefield,
Worst: Jungle Japes, Frigate Orpheon, Castle Siege