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 finishers, 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 Marth player may counterpick Final destination when playing against Falco, as most of the time they work very well with Marth's playstyle and allows for uninterrupted combos. 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.