Character Counterpick
Counterpicking is the act of choosing a character in order to have the advantage on a character that your opponent has chosen. A good example of this is picking Fox against a Jigglypuff. Fox is excellent against Jiggs because Fox's up smash and up aerial are his best killers. Jigglypuff, being a light character, will die from said moves from as low as around 55%.
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 (stage) 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.
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.