Basically, a banned stage wil never be played ever, ever ever. In melee, stages that are banned include Icicle mountain, Flat Zone, Big Blue etc.
A counterpick is a different kind of stage, in a tournament, the victor is decided by a set of 3 (or 5 if its the finals) If you lose a match, you can then counterpick the next map, that is, you choose the next map you play. It can be any of the 6 neutrals (we're still using melee as an example) or one of the many maps designated "counterpick stages" examples of counterpick stages are Brinstar, Green Greens, Mute City, etc etc.