Stadiums provide an effect for both players while in play. Usually, the Stadium chosen will be one that disrupts the opponent, while your own deck is unaffected by, or even benefits from it.
Stadiums can be so impactful that they reverse a matchup entirely, but they are very fragile, as the opponent can play Windstorm, Tauros, or a Stadium of their own to remove yours. Scott (elaborated below) is a good way of winning this Stadium war.
If you are familiar with the matchup and know that your Stadiums are ineffective against your opponent, while theirs are effective against you, you will have to use your Stadiums as pseudo-Windstorms. Hold on to your Stadiums to remove your opponent's; don't play or discard them without a good reason.
In a matchup where both players are playing only one type of Stadium which is ineffective against both of them (e.g. a Rai-Eggs mirror where both sides are on Cursed Stone only), the first player to play a Stadium has an advantage, as that Stadium will stay in play and prevent the opponent from playing their own copies to thin their hand (for Steven's Advice) or deck (for Rocket's Admin.).
If your opponent has a Pokémon Tool that you want to remove, but also an ineffective Stadium currently in play, you are not obligated to discard the Stadium with Windstorm. Leave it there so they can't play other Stadiums with the same name.
This punishes Poké-Powers, but does not stop them.
Often played alongside Cessation Crystal, or instead of it (if the deck's main attacker is a Pokémon-ex and cannot equip Cessation Crystal).
This punishes Pokémon-ex.
Most Pokémon-ex have high enough HP to be affected by it. Some notable exceptions are Mew ex, Banette ex, and Rocket's Sneasel ex. The last two even seize the opportunity to play Desert Ruins themselves!
The Damping Sphere of Pokémon.
This mainly counters Double Rainbow Energy, Scramble Energy, and bounce Energies, reducing the total amount of Energy they provide while also color screwing.
This does not affect Special Energy that only provide 1 Energy, such as (δ) Rainbow Energy or Multi Energy.
This serves several creative purposes:
It gets rid of dead Bench-sitters who would otherwise become Pow fodder, like a Jirachi left over from the early game.
It can be used to discard your own damaged Pokémon (especially Pokémon-ex) in order to deny Prizes.
It weakens opposing attacks that rely on a large Bench, such as Nidoqueen's Power Lariat or Exeggutor δ's Delta Circle.
It loosens your opponent's grip on the game if they are playing a lot of lock/engine pieces that work from the Bench. Lunatone + Solrock is a good example, as it takes two Bench slots — if you Stump them and they choose to keep 2 attackers and Lunatone + Solrock in play, then you KO their Active attacker, they only have 1 attacker left and may find it hard to make a comeback from that point.
It can be used to discard Pokémon with when-played Powers, so you can play Pokémon Retriever and use them again. (But note that you will have to make space on your Bench, likely by discarding Giant Stump, in order to replay the Pokémon.)
Strange Cave is a bit like Rare Candy for Fossils, in that it acts as a wildcard Basic (like how Rare Candy acts as a wildcard Stage 1), and lets you play an Evolution without the Basic having been in play for 1 turn.
Decks that play Fossil Evolutions will use this to get the Stage 1 forms of those Pokémon out, instead of the intended method of evolving from Fossil Trainers/Pokémon.
Scott (see below) can search for both Strange Cave, and Elm/Celio for the Stage 1.
Scott is used to fight through attempts at removing your Stadiums. It typically searches up two Stadiums (which can be duplicates), providing a backup if the first one gets removed, along with a Supporter for the next turn. Rocket's Admin. is a good choice, as it disrupts your opponent's hand before you pass the turn with the second Stadium up. (But note that if they have Pidgeot and you didn't play Battle Frontier, they can just Quick Search for an answer to it.)
Against Stantler's Push Away, Scott can also be used to get two Supporters to force at least one through it.
You can thin your deck with Scott, albeit slowly, by searching up a useless Stadium/Supporter, plus TV Reporter or any Holon Supporter to discard the dead card next turn.
Scott is usually found in decks that lack CA-generating Poké-Powers. It is a +2, and can guarantee you a Supporter for the next turn (unless you get Admin-ed), while also giving you two Stadiums to throw at your opponent.
Crush Chance removes a Stadium.
Tauros can remove a Stadium if your opponent has you Trainer-locked and unable to play Windstorm or a counter Stadium. The prominent Trainer-locking decks (Manectric ex and Dragonite ex δ) use ex's, so they cannot use Cessation Crystal to stop Tauros from removing the Stadium. Crobat δ can, though.
Call for Family makes for a decent setup attack if you happen to draw Tauros in your starting hand.