Lonely - Must have only STAB moves and no one of its type in its party
e.g. If you had a Lonely Lucario, it can only learn fighting and steel type moves and no one of fighting and steel in the party
Adamant - Must keep the first 4 moves it knows once in party
e.g. If you have an Adamant Tyrunt, it must keep all moves throughout the game once you unfossilize it. Another example is an Adamant Zigzagoon. You can still teach it moves in the empty slots but once all the slots are full, you can't change any of the moves
Naughty - Never use Poke-Amie or Refresh and if not in Single battles, must attack partner(s) first and roll a d20 everytime you choose a move, if 1, the Naughty pokemon doesn't accept your command and you must choose a different move
e.g. If you have a Naughty Crobat in a Double Battle with your Hippowdon in, he'll have to attack your Hippowdon first before your opponents. Also, when that Crobat uses a move like Poison Fang and you wanna use it, and you get one on your d20, you must choose a different move it knows like Sky Attack. While if you rolled a 14, then you can use Poison Fang.
Brave - Once in battle, it can't be returned until fainted or a nature forces a Pokemon to swap with this one
e.g. If you had a Brave Braviary, lets say you're fighting a Ludicolo and your Talonflame went down and you wanna use flying types, it can't be returned to swap out with another pokemon like Greninja until Braviary faints. Although, if you had a lets say Docile Greninja and its your first tyrn and the Brave Braviary was in the front of the party, it'll be switched out with Greninja
Bold - Shall always be in the front of the party at all times
e.g. you have a Bold Magmortar, it shall always take the first slot of the party when it is in your party
Impish - It has to be a ghost or dark type, and if none of those types, it has to be released plus when you're switching out a pokemon, roll a d6, if 6, you must switch out with the Impish pokemon
e.g. If you have an Impish Banette in your party and your Talonflame is in the red for HPs and you wanna switch it out, and you roll a 6, you must switch out with Banette. While on a 4, you can switch out with anything
Lax - Cannot ever hold Items
e.g. lets say you love Charizard's mega evolution so you choose Charmander from Sycamore and if its a Lax Charmander, you can't use the Charizardite X on it once it fully evolves into Charizard
Relaxed - Must know a sleep move or yawn
e.g. Lets say the Snorlax you caught on Route 7 has the relaxed nature, it must know a self sleep move like rest or it can learn yawn and if it can't, you can't use it in battles until it does
Modest - Can only learn Attack moves that doesn't have a more than 50% chance of activating their second effect
e.g. A Modest Dusknoir can't learn moves like Will-O-Wisp since it is a status move, plus, it cant use Phantom Force thanks to making the opponent waist some PP while also being a Priority move (correct me if I am wrong)
Mild - No Restrictions
e.g. A Mild Greninja does nothing to hinder gameplay and can be used almost as normally as in a regular play through
Rash - Can only learn attack moves and Defence shall always be its lowest stat
e.g. A Rash Talonflame can only learn attack moves and it must have a lower defence than its SpD or HP
Quiet - Can't attack sleeping pokemon and can never learn noise moves
e.g. If you have a Quiet Xerneas, it'll never wanna attack sleeping pokemon like a resting Snorlax plus it won't wanna learn moves that deal with sound like Uproar. Also you have to release Quiet Whismurs, Loudreds, and Explouds since they can never be quiet
Calm - Can never learn Frustration or any of the rage moves and is always a last resort
e.g. A Calm Espeon can only be used when all your party members have fainted plus it can't learn Frustration or moves like Rage or Bide
Gentle - Can only learn moves with a power of 60 or less
e.g. A Gentle Scraggy cannot learn moves like High Jump Kick which is above 60 power while it can learn moves like Feint Attack with a power lower than 60
Careful - Shall never make the Final Blow and when in red HPs, shall always be returned
e.g. A Careful Flabebe can never use a move like Fairy Wind on those pokemon with HPs in the yellow or red and plus when its HPs is in the red, it has to switch out
Sassy - If capable, shall always know a fire type move and shall be switched into the first time you switch your pokemon during a battle, overriding Impish
e.g. Lets say you have a Sassy Shiny Skiddo and your Bold Magmortar is dying, so you wanna switch. Switch in your Sassy Shiny Skiddo if its your first pokemon returned, even if there is an Impish Banette in your party that rolled a 6
Timid - Once hit, it shall always switch out no matter what
e.g. You have a Timid Talonflame in a battle with a Ludicolo. Once that Ludicolo hits you with lets say Water Gun, the next turn, Talonflame has to be returned even if it didn't faint
Hasty - Shall always know a Priority move and if the pokemon can't learn one till Lv40, you must release it
e.g. If you have a Hasty Scizor, it must know a move like Bullet Punch before LV40. Maybe you might have a Hasty Ferroseed which has to be released thanks to it not able to learn a Priority move before LV40
Jolly - Shall always have a Happiness above 150 and can only be used in the daytime
e.g. Lets say you just got a Jolly Happiny, it has to have happiness over 150 at all times and can only be used when its daytime in-game (so, in M,UM, it is only used in Nightime in real life)
Naive - Can't learn any Psychic type moves and Clever Contest moves and can't level above LV50
e.g. If you have a Naive Dragonair, it can't ever learn Psychic moves and Clever contest moves (like Fly). Plus because of level restrictions, it can never evolve into Dragonite
Hardy - Shall always have a stay boosting move and if there is a hazardous weather, it'll stay in battle no matter what
e.g. A Hardy Hoppip you have has to have a stat boisting move like Cotton Guard and even if there is a Sandstorm going on, it'll stay out in battle
Docile - Shall always be switched in during first turn of battle
e.g. Lets say you got a Docile Diggersby, in every first turn, if not already out, has to be sent out and it trumps all other natures while doing so
Bashful - Can only be used if only one left overriding calm
e.g. You have A Bashful Heliolisk, it can only be used as last resort no matter if you have a pokemon with calm or not
Quirky - roll a d4 everytime you attack, your roll will correspond to a move your Pokemon knows (top to bottom)
e.g. You have a Quirky Sawk with the moves Knock Off, Brick Break, High Jump Kick, and Bulk Up, if you roll a 4, it uses Bulk Up. 3 HJK, 2, Brick Break and so on
Serious - Shall always have a GOOD moves and a held item
e.g. A Serious Froslass shall have a moveset that can be use to its max (like Destiny Bond mixed with Hail and Blizzard, plus it knows Double Team) and it has to have a held item it can use (like the Icy Rock)