Character counterpicks may be made after a lost 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 counters Jigglypuff because his up smash and up aerial are his best KO moves, and Jigglypuff will get KOed by said moves at around 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 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.