Right... this is gonna be a long post:
Let's start with the effects on types due to Fairy. Firstly, I don't think Dragon types are going to lose popularity. Instead, we'll just see less Outrage and 'Dragmag' strats. Dragons can still blast a lot of the better fairies with their coverage moves with ease, indeed, I am convinced the only reason Fairy isn't weak to Fire is because Fairy, as a balance plaster for Dragon, can't be weak to Fire Blasts. We'll still see a lot of Dragon Dancers and Wallbreaker dragons, just less OUTRAGE SPAM AFTER MAGNEZONE KILLS A STEEL.
Bulkier Fire types are going to gain some more popularity. Moltres may even make low OU instead of UU/BL. They probobly won't raise that much due to Stealth Rock, but combined with the Drizzle nerf, and Fire may just stop being called an awful defensive type... especially with 6 resists.
On the other hand, Poison types are not going to suddenly shoot up just because they finally have a type they hit super-effective that isn't most often paired with Poison as a secondary type, or has a huge amount of better types to abuse on it. Poison is still an awful offensive type, although it's even better defensively now.
Steel types should be relatively unaffected overall, their defense got nerfed, the offense buffed. Most carried Gyro Ball anyway. We may just see more offensive Metagross, which can set up Agility on Fairies.
We may also see less U-Turn spamming, with another Bug Resist in the game. Except Scizor is also known for Bullet Punch.
Now; onto the fairies themselves:
Gardevoir -
OK, this thing is gonna jump. It's currently NU, and not even the premier psychic. However, with the Fairy-type added to it [Especially since it's only neutral to Poison, so only gains a Steel weakness while losing Dark and Bug weaknesses], and it's Mega Form as well, Gardevoir is probobly going to jump in useage to at least UU. It's one of the strongest fairy-types in the game, and has quite a good shot of being the main Fairy in OU, with maybe 3 competitiors.
Togekiss -
This thing is ALREADY hugely annoying, and it's just gained a bunch more resistances [And weaknesses], as well as tadeing one immunity for another. Togekiss will likely remain a huge factor in OU, and Nasty Plot sets will likly become more common thanks to STAB Fairy Special attacks on top of Air Slashes. Probobly competeing with Gardevoir for #1 fairy.
Mawile -
This thing has a shot at getting into OU simply due to it's Mega Form's base ATK, Huge Power and Sucker Punch. Less so for the Fairy-type, although Steel/Fairy is an excellent type in general, with 2 immunities, and only 2 weaknesses. [Fire and Ground]. STAB Huge Power Play Rough is scary though.
Azumarill -
Speaking of Huge Power... Azumarill's Choice Band set suffered from being a one-trick pony in Gen 5. Aqua Jet was it's main move 90% of the time. It even struggled to fill it's movepool. Well, now, it's got Play Rough to use on predicted switches to types that can take Water-type attacks, such as... Dragons? Water and Fairy is excellent type coverage as well. Unless I'm mistaken and have forgotten something, only Shedinja, Ferrothorn, Tentacruel and Emploeon resist it.
Klefki -
Although it likly won't make OU, I can see this thing being a prime annoyer in UU, with Tormet+Sub with Prankster. It would be like TormentTran... but significantly weaker. Also Prankster Swagger, Prankster Spikes, Prankster Screens... oh, and Foul Play just in case you want to SD on it or something.
The other Fairies:
Most of them are pretty bad. Xerneas is obviously going to Ubers, so I won't speak about it. Aside from maybe Florges and Sylveon, who would be solid UU/RU special attackers, and Whimsicott, who will pop up like it did in Gen 5 occasionally, the rest just lack the stats and would likly be NU, and Fairy lacks a great movepool as well to make up for the stats.
Note; there may still be more Fairy-types that are yet undiscovered to have changed; most of these are legends, like the Lake Trio, BST600 Pixies [Celebi, Jirachi, Manaphy and Victini in particular come to mind, Mew less so.], and possibly Cresselia [As Fairy > Dark]