Specific move
Lickitung - Lickilicky
-levelled up when knowing Rollout
Aipom - Ambipom
-levelled up when knowing Double Hit
Mime Jr. - Mr. Mime
-levelled up when knowing Mimic
Eeveelutions
Eevee - Leafeon
-levelled up near Mossy Rock
Eevee - Glaceon
-levelled up near Icy Rock
Eevee - Sylveon
-levelled up w/ 2 hearts in Pokemon Amie
Specific time
Amura - Aurorus
-levelled up past level 39 at night
Tyrunt - Tyrantum
-levelled up past level 39 during the day
Specific held item (level up)
Gilgar - Gliscor
-levelled while holding Razor Fang at night
Sneasel - Weavile
-levelled while holding Razor Claw at night
Happiny - Chansey
-levelled up while holding Oval Stone
Specific trade requirements
Shelmet - Accelgor
Karrablast > Escavalier
-when they are both traded for each other
Clamperl - Gorebyss
-traded while holding a DeepSeaScale
Clamperl - Huntail
-traded while holding a DeepSeaTooth
Seadra - Kingdra
-traded while holding a Dragon Scale
Porygon - Porygon2 > Porygon-Z
-traded while holding an Upgrade
-traded while holding a Dubious Disc
Electabuzz - Electivire
-traded while holding an Electiriser
Magmar - Magmortar
-traded while holding a Magmariser
Feebas - Milotic
-traded while holding a Prism Scale (Gen V+)
Rhydon - Rhyperior
-traded while holding a Protector
Dusclops - Dusknoir
-traded while holding a Reaper Cloth
Spiritizee - Aromatisee
-traded while holding a Sachet
Swirlix - Slurpuff
-traded while holding some Whipped Cream
Misc. requirements
Mantyke - Mantine
-levelled up with a Remoraid in the party
Pancham - Pangoro
-levelled up past level 32 with a Dark type in the party
Feebas - Milotic
-levelled up when beauty condition maxed (prior to Gen V)
Silgoo - Goodra
-levelled up past level 50 when it's naturally raining in the overworld
Tyrouge - Hitmontop, Hitmonchan, Hitmonlee
-levelled up past level 20 if:
~Attack = Defence (Hitmontop)
~Attack < Defence (Hitmonchan)
~Attack > Defence (Hitmonlee)
Inkay - Malamar
-levelled up past level 30 while holding your 2/3DS upside down
Source