The first generation of Pokemon games, commonly referred to as "RBY" but actually consisting of Red, Blue, Yellow, and Stadium*, was the first competitive system around which Pokemon battling was formed.
By the time GSC was released in the United States (and in Japan), and with it the limited ability to transfer Pokemon from these games back to the RBY games, the RBY metagame was already established; when GSC came out, the focus shifted away from RBY, anyway.
With these new Pokemon games came the ability for Pokemon that exist in the RBY generation to learn moves in Gold/Silver/Crystal which exist in the RBY generation but are not learnable by those specific Pokemon in the RBY generation. These techniques on these Pokemon are called "tradebacks."
Besides moves which are learned in the GSC games themselves, whether by level, breeding, or TM, there are also special "give-away" Pokemon that the New York Pokemon Center traded to people during special events. These Pokemon are ordinary, with the addition of a new move. Because of the nature of this "medium," you can't have a Pokemon with both an NYPC move and a move learned through breeding.
However, due to the situation as described earlier with the presence of an established RBY metagame, most battlers (and thus, the competitive community overall/as a whole) chose not to standardize these new tradeback moves into common play. So, as it exists today, competitive RBY battlers generally prefer not to play with tradebacks ("Pure RBY").
In any case, tradebacks and the tradeback metagame remain an interesting and notable aspect of competitive Pokemon, especially since there are those who enjoy playing RBY with tradebacks (and do so) to this day.
*This is mentioned to clarify why Surf Pikachu isn't considered a "tradeback."
Nidoking, Clefable, Persian, Hypno, Kingler, and Tangela learn Amnesia (all but Hypno via breeding). If you know anything about RBY you know that Amnesia doubles both special offense and special defense.
Benefiting most are Nidoking and Clefable, both of whom can learn any of Thunder, Thunderbolt, Blizzard, and Ice Beam. Kingler and Tangela can only learn STAB (and near-STAB, in the case of Kingler and the Ice-type) special attacks, but become the first and only Pokemon to learn both Swords Dance and Amnesia. Persian and Hypno learn Thunderbolt, then BubbleBeam and Psychic respectively, for a fairly varied offense that's still helpless to Exeggutor.
We have a bunch of new Lovely Kiss users out of the NYPC Valentine's Day releases, as well as several Pokemon which learn Hypnosis through breeding; Nidoking/Nidoqueen and Snorlax (and Victreebel, though it's only useful as a novelty technique that's the same accuracy but with 10 PP instead of 15) learn the former, while Ninetales, Golduck, Persian, Rapidash, and Mr. Mime are all now capable of the latter. Oh yeah, and Raichu can Sing. Adding all these new good UU-ish sleepers definitely throws a wrench into the metagame's mix.
Ninetales becomes similar to Gengar with its Hypnosis/Confuse Ray combination, and a more solid lead. Persian beats out the ghost for the title of fastest sleeper, and becomes a more solid lead as well. Too bad Golduck can't legally learn both Amnesia and Hypnosis; looks like Poliwrath's still got it. All the Hypnosis users are pretty fast and could definitely be someone's potential lead. Sing makes an awesome filler for Raichu, though Body Slam still rocks.
Swords Dance makes Marowak much better and moves it up a tier, though it's still outclassed in every stat and in the rest of its movepool by Sandslash. Swords Dances makes Seaking something to actually be wary of, as it can now confuse with Supersonic and/or Agility up, and attempt to sweep; it moves up two tiers to the same one as Marowak, just below borderline.
Several Pokemon finally get the move they always wanted. Hitmonchan scores Hi Jump Kick, moving up a tier although still probably a step behind Hitmonlee. Vileplume finally scores Razor Leaf, although still probably a step behind Victreebel. And Aerodactyl can finally stand up Rhydon and Gengar with a solid, damaging Ground-type Earthquake attack.
The Water-types Starmie and more notably Tentacruel both now have access to Confuse Ray, allowing Starmie a chance at parafusion and Wrap Tentacruel a chance at even more cruelty. Tauros is even more unpredictably damaging now with Surf, which does a tremendous amount of damage to Golem and Rhydon and coupled with just one Body Slam may often result in a KO. Cloyster learns Screech--followed by Explosion, that's one clam you need to keep both eyes on (and one that's not as easily countered as Electrode).
Alakazam and Hypno learn the Ice-type and Electric-type elemental punches; Alakazam will nearly 2HKO Starmie with high damage ThunderPunch.
Alakazam, Cloyster, Hypno, Starmie, Electabuzz, and Magmar all now learn Barrier through breeding. These Pokemon could all learn Reflect previously, it's true, but the ability to use such a technique several times can sometimes come in handy. (Some of these Pokemon may be better off with another move that's incompatible with this, however, such as Hypno's Amnesia).
Fearow, Golbat, Rapidash, Farfetch'd (!), Dodrio, and Tauros now may sport Quick Attack. I personally love this attack, so, yeah. I'd expect it could become standard filler on Dodrio, if not for the addition of Low Kick on Dodrio, a 50 Base Power Fighting-type attack that takes off a little less than 20% to Golem and Rhydon (which definitely takes the cake).
Generally, Hazers are far from what they used to be (which wasn't that much anyway, unfortunately), but in this tradeback metagame with all these Amnesia/Swords Dance/Barrier users, maybe it'll have more use. Try it on Blastoise, Poliwrath, Tentacruel, Dodrio (!), Muk, Gengar, Kingler, Seadra, Seaking, Omastar, or Dragonite. Many of those Pokemon have stat-up moves that could conflict with Haze, but I can see definite use in Gengar and Dragonite.
Primeape, Jynx, and Electabuzz all learn Meditate, but that's tough to pull off. It'd be interesting to see how this could work on Jynx (not at all, probably) which has the best chance of stalling out of these, but it would require the use of two slots. (Speaking of Electabuzz, that and Magmar can now Karate Chop for solid 50x2 always critical hitting Normal-type attacks.) Dugtrio can now Screech, which I see as just another opportunity to use prediction to encourage effective use (or you can just skip it and use critical hit Earthquake every turn for the same effect, since it is Dugtrio).
Scyther learns Counter, making it the second Flying-type after Charizard to have access to this rocking technique, and it may find some use for the same reasons Charizard may value Counter (e.g. countering physical Snorlax with prediction). Speaking of Charizard, that Pokemon can now use Rock Slide, which could probably come in handy in UU.
Venusaur, Parasect, Alakazam, Hypno, Machamp, Scyther, and Dragonite can all help defend against special attacks with Light Screen, but I'm not sure if this could become standard on any of those. I'd look toward Machamp, Scyther, and Venusaur, however.
The Electric-types Magneton and Electrode may now use Agility, which could work well if they're paralyzed and wish to exploit the speed boost glitch.
Raticate and Sandslash can also Counter now! Probably more useful on the former than the latter, however, since Sandslash already has a nice movepool and isn't likely to be hit with non-Earthquake physical attacks anyway.
Some other loose ends are Flamethrower Moltres, which is probably not usually preferred to Fire Blast but is definitely worthy of note, and Hydro Pump Dragonite (?).
And the Eeveelutions can learn Growth via NYPC, which is awesome and could help something like Jolteon to overcome Chansey, or Alakazam maybe!
Organized out of a discussion with contributions from many other Smogon members, notably Amazing Ampharos, GotMiltank, JMC, Warthog, Altmer, Kikuichimonji, AJC, and with the refinement of others who are able to provide constructive criticism on this summary.