Character counterpicks may be made after losing a match in a tournament ruleset. For example, after losing to a Jigglypuff player in Super Smash Bros. Melee, a smasher might counterpick with Fox. Fox soft counters Jigglypuff because although he is more susceptible to Rest setups than most other characters, he in return possesses faster movement options and equally as powerful options to KO Jigglypuff; his up smash and up aerial, which are his best KO moves, will take out Jigglypuff at as low as 55% damage 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 Melee, a Fox player may counterpick Pokémon Stadium against various characters, as the stage is large enough for said character to laser camp and exploit fast movement options, and the low ceiling on the stage greatly benefits Fox's vertical finishers. Alternatively, a Marth player may counterpick Final Destination against fastfallers, as the lack of platforms for the enemy to escape pressure allows Marth to chain grab or tech chase them without fail. Stage counterpicks are often regulated by Dave's stupid rule.