Why nobody mentioned these two guys yet?
Celebi is used mainly for its ability to check or outright counter rain teams, but Celebi has other uses, like countering top-threats like Keldeo and Sheer Force Landorus, or sweeping rain teams and weakened sand teams. Celebi is also the best counter to Breloom. Sun teams hamper Celebi's effectiveness, but when paired with the proper weather support, not even them are a problem for Celebi. This Pokémon is also useful to keep Fighting-types at bay, as Celebi can deal with most of them.
The specially defensive set is one of the most used sets, and for good reason. This is the set that wall many top-threats like the aforementioned Keldeo and Landorus. It is able to wall any rain team that doesn't pack Tornadus or Scizor, two of the most effective ways to get rid of Celebi on rain teams. However, Celebi can use an offensive Nasty Plot set to sweep the same rain teams instead of walling them. With the proper support, this set can also sweep weakened sand teams, weatherless teams, and can deal with some threats of sun teams.
Peharps what makes Celebi able to fill these roles, is its stats, combined with its movepool and unique typing. 100 base stats aren't what they used to be, but a Pokémon with 100 base stats is still very useful. Celebi is one of the few threats that has sufficient defenses to tank those rain-boosted Hydro Pumps from the likes of Keldeo and Politoed, of course, the Psychic/Grass typing helps here. Speaking of which, Celebi's typing is extremely underrated, just because it gives an extremely unfortunate number of weakness. Celebi is weak to U-Turn, weak to Pursuit, weak to Ice Shard, weak to Fire-type attacks, and is weak to Hurricane. The weakness to Hurricane and U-Turn is the reason why Celebi couldn't keep up on a metagame where Genesect dominated, but it was still very useful to counter Water-types, but when Tornadus-T was dominating, Celebi was still put on a though position, though usually once Tornadus-T was gone, Celebi could wall rain teams eternally. Well, these weakness are certainly unfortunate. However, nowadays Hurricane is a rare move to be used, nobody uses Poison-type attacks, both Pursuit and U-Turn are very predictable, and Celebi can tank many Ice-type attacks. So, very few of its weakness are actually relevant. Psychic/Grass is an underrated typing because it also gives awesome resistances to Fighting, Psychic, Water, Ground, and Electric. These resistances are what give Celebi the ability to destroy Fighting-types and Water-types, as well as Ground-types. Lastly, Celebi has a rather impressive movepool for a Grass-type. Apart from its STAB moves, Celebi has Hidden Power coverage (which is usually sufficient) and Earth Power, to smash any Heatran that thougth that they could counter Celebi. While Celebi's offensive movepool is not that great, it contains every option that Celebi needs for coverage, and it even has U-Turn to mantain momentum, and its defensive movepool has amazing support options like Perish Song (an extremely useful move that unfortunately has shallow distribution), Thunder Wave, Baton Pass, and even Leech Seed.
If you want to deal with Celebi, one of the best ways is Tyranitar. Watch out for Nasty Plot sets, as they can effectively deal with Tyranitar. However, defensive sets have problems as they are extremely prone to being Pursuit trapped, unless Celebi carry Baton Pass. Peharps this is one of the reasons why Tyranitar is so omnipresent, as Celebi can wall threats like Sheer Force Landorus and Keldeo, and Tyranitar can get rid of most their counters. This is a proof that Celebi has contributed to shape the metagame. The Nasty Plot set can get past Celebi's common counters and checks, like Tyranitar, Scizor, and Heatran, and thus, the best way to deal with Celebi on that case is to pack a powerful Pokémon that can outpeed and hit with a move that Celebi is weak. There are very few Pokémon nowadays that fit this description. Some of them are Terrakion with X-Scissor and physical Landorus (with U-Turn), but they are rare.
Keldeo was introduced as one of the threats that every team should prepare for. It was not too late until people realized the mighty power of Hydro Pump. While Keldeo can sweep late-game with its Scarf set, its Specs and Calm Mind sets bring absolute power to the table. When not boosted by rain, these sets are already though to deal with, but if rain-boosted, only the bulkiest of resists is able to effectively deal with Keldeo. This is the reason why people started to run things like Amoongus, Jellicent, and Celebi, which is a proof that Keldeo is shaping the metagame.
The most used set is Choice Scarf, as it is very hard to find something that can outspeed it and take repeated hits when boosted by rain. This forces people to use defensive checks and counters to deal with Tyranitar. However, this is not the only set that Keldeo can run. Choice Specs absolutely kick asses, and hits hard right off the bat. It is very hard to find something that can tank hits from this set. Calm Mind unfortunately has to waste time setting up, and has lesser coverage, but as it can switch moves, this set has peharps more sweeping potential. The SubCM set is rare, but it is so dangerous... It can also deal with its normal checks and counters. The recent Expert Belt set is also dangerous, as it can effectively deal with Keldeo's counters. Thougth that your Celebi was a foolproof counter to your Keldeo? Met a Hidden Power Bug on your face!
What peharps make Keldeo have impact, is its Water/Fighting typing, combined with its stats. Keldeo has the same stats as Terrakion; only its Special Attack and Attack are inverted. It sits at an awesome speed tier, easily outspeeding threats like Landorus, Thundurus-T, and many Dragons like Dragonite and Salamence. Fighting gives Keldeo a resistance to Stealth Rock, as well as an awesome STAB that lets Keldeo deal with three of the biggest enemies to Water-types around; Ferrothorn and the pink blobs. Also, all you know that Fighting is an awesome STAB move to spam, as even the few Fighting-type resists are hard pressed to tank repeated Fighting-type blows, although Keldeo's icing on the cake is its Water-type STAB. Thanks to Keldeo's awesome Special Attack and rain, Hydro Pump is so powerful that simply resisting Hydro Pump is not sufficient, the pokémon should resist Hydro Pump AND have the defensive stats to tank repeated Hydro Pumps. While Keldeo has a shallow movepool, it gets an interesting move; Secret Sword. It is not as powerful as Close Combat, or even Focus Blast, but it hits enemies on their physical Defense. This means that Keldeo is the only special sweeper on the game that can easily bypass both Chansey and Blissey, without any prior setup! This peharps has contributed to shape the metagame, as thanks to Keldeo, Blissey and Chansey are no longer useful as special walls. Also, despite Keldeo's shallow movepool, Hydro Pump is so powerful that it can often compensate for that. However, it is certainly a shame that Keldeo gets to the point to miss things that other Water-types do not miss, like Ice Beam. Also, Water/Fighting is a rather awesome typing defensively speaking. It resists Stealth Rock, resists Ice Shard and Bullet Punch, is not weak to Mach Punch, and resists U-Turn and both Water and Fire. Keldeo has a respectable bulk, but you shouldn't be counting on it to tank hits. However, with that bulk, Keldeo can take resisted hits well, and even some weaker super-effective hits like Starmie's Thunderbolt.
It is very hard to deal with this thing. Usually the only way to deal with Keldeo is using more defensive Pokémon like Latias, Amoongus, Celebi, and Jellicent, all which can stop Keldeo on its tracks. However, all of them are weak to Tyranitar, and thus even then are not a reliable way to stop Keldeo. Toxicroak also usually counters Keldeo, and unlike the aforementioned counters, Toxicroak does not fear Keldeo at all. However, it needs rain to function, so it is an unreliable Keldeo counter. When Keldeo is not using a Choice Scarf, however, it is easier to deal with. Revenge killers like Latios, Alakazam, and physical Landorus under sand, easily beat Keldeo. The problem is that the most common Keldeo set is Choice Scarf, which easily destroy offensive threats. All things combined, Keldeo is certainly one of the titans from the metagame, as it helped to shape it and make awesome cores with other Pokémon.