basically, each tournament creates a list of stages that are legal to play on during the tournament. this list is divided into neutral stages and (possible)counterpicks.
neutral stages usually include final destination, fountain of dreams, battlefield, yoshi's story and dreamland 64, these stages (neutral stages) are the stages that are available for the first match of a set. (i won't go into stagestriking right now)
the player who loses a match in a multiple match set (like best of 3 or best of 5) is now allowed to pick the stage for the next match in that set, hence counterpicking. he now may chose between all the neutrals and also all the counterpicks and chose the stage he feels the most comfortable with. if "daves stupid rule" is in effect, he however is only allowed to pick a stage that he did not win on in this set of matches.
example:
we play a best of 3 set and via stagestriking or whatever we play the first match on battlefield. you win. i can now chose the next stage that we will be playing on and i chose pokemon stadium (a usual counterpick stage). i win this match and that gives you the right to chose the stage, however since we play with daves stupid rule in effect, you can not pick battlefield. you can take me to all the other legal stages of the tournament, both neutrals and counterpicks.
what stages are banned is hard to answer, because for a tournament, the organizers decide which stages are legal. (should be mentioned in the thread of your tournament)
if you are referring to the other meaning of banned stages, that would also be hard to answer, since each player can ban a stage per set, which makes it entirely dependant on who your oponent is and what stages you hate most.