Suppose that my opponent plays an Aethering:
, and I can't counter it so it arrives on the battlefield. How can I remove that card from the battlefield? I'm playing Standard Constructed.
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Removing Aetherling is useful, but not always necessary. The key thing is: how can you use the cards you have in your deck to deal with the threat?
Turn off its abilities
Aetherling is only powerful because of its activated abilities (in particular, the unblockable and bounce abilities). Pithing Needle will turn it into a vanilla 4/5, which really isn't a very powerful creature at this stage in the game. Which is a pretty good deal if you're stopping your opponent's best win condition or hosing multiple copies of it.
While an Aetherling can do significant damage every turn, it's not actually the fastest kill card in Standard. When the Aetherling comes down, look at the board position — are you better off trying to kill it or just killing your opponent?
If you plan to race, focus your removal on swinging the game's tempo. Use it to get rid of blockers (including Aetherling, if only for a turn) so your guys can get in more damage. You can also use instant-speed removal like a Fog against the Aetherling.
Life gain can be very useful to keep you afloat. If you're playing Bant or BUG in a control mirror against UWR or Esper, for instance, you'll likely have more life and more ways to gain life than your opponent. Even if you can't hold out forever, you can try to use your life gain to force your opponent to commit more than just U (for the unblockability) to the Aetherling each turn in order to kill you quickly. At which point you can take advantage of the next tactic...
Overload your opponent's mana
Aetherling is only survivable as long as your opponent has access to blue mana. Watch you opponent's mana usage. If you have more ways to kill the Aetherling than he has blue mana, it's dead. You may need to spend more than one spell to do this, but it's better than losing the game altogether.
The easiest example is right after he casts it. Say you're playing Jund and your opponent just cast Aetherling with one Island in reserve to flicker it. You've got Liliana of the Veil and Putrefy in hand. Play Liliana's edict. When they put the Aetherling's exile trigger on the stack (spending their last U to do it), Putrefy the Aetherling in response.
Prevent it from attacking
Aetherling is annoying to get rid of, but it's still just a creature that has to attack you to end the game. Sphere of Safety and Stormtide Leviathan are the only such cards in DGM Standard, but if you have access to a wider card pool you can play cards like Moat, Blazing Archon, or Ensnaring Bridge (your opponent could pump down the Aetherling to allow it to attack, and then pump it back up to allow it to do damage, but that costs a ton of mana, so it's not exactly a problem for you).
"TurboFog" decks are also very effective at blocking out creature damage, even from nasty creatures they can't kill outright.
Debtor's Pulpit was used in Pro Tour Dragon's Maze to lock down Aetherling. It's a CMC-5 aura that gives the enchanted land the ability to tap a target creature. You can tap Aetherling at the beginning of combat. It they blink it, it can't attack during that combat anyway.
As it has already been said, but I will second Pithing Needle. You don't need anything more than that, as it can turn Aetherling into a 6 mana cost vanilla 4/5 creature. You don't even need to let the Aetherling hit the table; On the first turn you can drop Pithing Needle naming Aetherling and he's taken care of. Plus there are many other targets in Standard, like Planeswalkers and Olivia Volarden.
To be honest you can't do anything if they 3 or mana up as they resolve it. However you can stall for the win. Remember this card has 2 downsides it cost mana to keep it alive and it cost mana to end the game. Keep your removal spell to create a fog effect and keep swinging. If he blinks his creature its fine it just means he loses 1 attack step