Celebi
This is a bit long; I think I got carried away. If anyone could make this more concise, that would be great (if not, I will try later).
Blissey is a great initial switch-in to Celebi. It has Toxic to force Celebi to switch out, Natural Cure to eliminate paralysis from Thunder Wave, and Softboiled to recover any damage incurred in the exchange. Blissey only struggles with the occasional Leech Seed set or Baton Pass variant. Snorlax functions similarly to Blissey as a counter, trading bulk and reliable recovery for immediate attacking power. Offensive sets can threaten Celebi with their Choice Band-boosted STAB attacks or Pursuit, while defensive sets can set up with Curse. Curse-less defensive sets will have to rely on PP stall and/or Body Slam paralyses to beat Celebi, though. While it doesn't necessarily take a dedicated special wall to beat Celebi, keep in mind that most of Celebi's other counters and checks are crippled by Thunder Wave. Togekiss completely walls Celebi, but because it is especially reliant on Speed, it must avoid switching into Thunder Wave at all costs. Dragonite is in the same boat; additionally, it should take care to avoid the rare Hidden Power Ice.
Many of Celebi's potential checks and counters must also take care to avoid being surprised by super effective coverage moves. Heatran is a reliable answer to most Celebi sets, but a Nasty Plot-boosted Earth Power will OHKO even ShucaTran. Skarmory can withstand repeated Hidden Power Fires from a defensive Celebi, but hits boosted by either Life Orb or Nasty Plot will cause severe damage, if not an immediate KO. Crobat is another decent check, but switching into Psychic will severely damage it. Gengar operates similarly, except it's even worse off: Psychic actually OHKOes it, while Celebi can be EVed to survive Shadow Ball. Infernape, cursed with weaknesses to both Earth Power and Psychic, must exercise even more caution when switching into Celebi but is an effective check otherwise.
One of Celebi's most easily exploitable weaknesses is to U-turn. In addition to Crobat and Infernape, Flygon and Jirachi are capable of checking Celebi through U-turn, although both will get worn down if repeatedly switching into Grass Knot/Energy Ball and Hidden Power Fire/Earth Power, respectively. If they are Choice Scarfed rather than Banded, U-turn will not OHKO, either; a healthy Celebi has the option of staying in and using Recover, though Jirachi can take advantage of this and use Trick. Scizor, the premier user of U-turn in OU, should be extremely careful to avoid Hidden Power Fire, which will KO with residual damage. Its U-turn is the strongest and most punishing, however, and it can use Pursuit to heavily damage a retreating Celebi. Additionally, Choice Scarf variants can earn a surprise KO on a complacent Celebi staying into use Hidden Power Fire. Gliscor can beat most defensive Celebi by using Taunt to prevent Recover and then using U-turn; however, Leaf Storm from offensive variants will deal heavy damage to it. Azelf can switch in on anything that isn't Leaf Storm or Thunder Wave and threaten with a Choice Band boosted U-turn—if it hasn't used Explosion turn 1, anyway.
Celebi's other most commonly exploited weakness is to Pursuit. Tyranitar can survive anything up to a Specs Leaf Storm from Celebi and force it to predict between Crunch and Pursuit, although a wrong prediction can be as fatal to it as it is to Celebi. Weavile works in a similar fashion: it's a better switch-in to Grass Knot than Tyranitar, but worse for most of Celebi's other attacks. While Metagross is significantly weaker against Celebi—its Pursuit lacks STAB and its STAB of choice hits Celebi only neutrally—, unlike Tyranitar and Weavile, it can afford to mispredict a Celebi switch-out, as its bulk lets it survive a few Hidden Power Fires or Earth Powers. Heracross is not as competitively viable as OU's more conventional Pursuit users, but it can withstand a Hidden Power Fire or Psychic thanks to its respectable special bulk, and it can actually switch into Celebi's Thunder Wave and benefit from it thanks to Guts.
There remain a few other unmentioned checks to Celebi, but most are situational or set specific. Most Celebi carry Hidden Power Fire, but the ones that don't are vulnerable to the aforementioned Scizor, as well as Abomasnow and Forretress. If Celebi lacks Calm Mind and has Grass Knot as its STAB of choice, Rotom-A can force it out with the threat of Shadow Ball. Jirachi can use Body Slam in conjunction with Iron Head to flinch Celebi to death but will be rendered useless should Celebi hit it with Thunder Wave. Jirachi can also freely set up Calm Minds on most defensive Celebi; however, it will lose to an opposing Calm Mind set with super effective coverage and be forced out by sets with Perish Song. Roserade can set up entry hazards on defensive Celebi, but it has no way of beating it unless it carries Sludge Bomb. Funnily enough, Celebi can also check itself through using a slower U-turn to another check, using Perish Song to force its doppelganger out, or setting up with Calm Mind. Naturally, it should also beware these same strategies itself.