Counterpicking is the act of choosing a character, often a counter, in order to have the advantage on a character that your opponent has chosen. A counterpick 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 Fox's up smash and up aerial are his best killers. Jigglypuff will die from said moves at around 55% since it is a lighter character.
Stage Counterpick
Another form of counterpicking is stage counterpicking. This is mostly done based on the character that someone has chosen. For example, a Marth player will often pick the small Yoshi's Story because one of Marth's greatest shortcomings is his inability to fire any projectiles and his defense against them. This small stage will give characters that have projectiles less distance to fire them. This stage also favors non-projectile characters because the platforms are all close together, allowing non-projectile characters to close the gap much easier/faster. This stage also allows Marth to stand on the main platform, and use his forward smash to either side (having the tip of the sword connect with any characters on the bottom two platforms), giving him a huge advantage as the sweetspot of his sword reaches all of the bottom platforms.
Stage counterpicking can also mean that a player will choose a stage that is not suited well for his opponent's character. A good example of this can be seen when a player chooses Mute City when playing against Falco. When the platform is flying through the air, the ledges are not able to be grabbed and coupled with Falco's short-distance Up B move and predictable Forward B, it presents unfavorable conditions for a Falco trying to return to the stage after being knocked off. Stage counterpicks are often regulated by Dave's Stupid Rule.