Happiness

One of the less important game mechanics that is fully understandable without the use of complex equations and such is Happiness. It isn't really the most helpful game mechanic in battle unless you're speaking of the moves Return/Frustration, but it is a game mechanic anyhow. The other use for happiness would be for evolution purposes, which can obviously be an important factor for stats. Anyway, lets start off with some basics.

Logic would suggest that when you catch a Pokémon, it's not going to like you, but it's not going to hate you either. There's a way to increase the Happiness of a Pokémon that you've caught immediately as well, and that's by using a Luxury Ball instead of a regular Pokeball (or any other ball for that matter). Of course, you cant use Luxury Balls on hatched Pokémon, so that shortcut is kinda useless if you are a breeder. There are lots of other ways to influence happiness however, so it's not that important.

Happiness is actually as easy to influence as eating a piece of toast or turning a doorknob, or whatever other comparisons you can think of. The things that will raise happiness are plentiful and each raise different values, with the values changing based upon how happy your Pokémon is already

Sword & Shield

ActionCurrent Friendship
0-99100-159160+
Level Up+1+20
Faint from Pokémon less than 30 levels higher-1-1-1
Faint from Pokémon more than 30 levels higher-5-5-10
Battle Gym Leader or Champion+5+30
Walking (50% chance every 128 steps)+1+10
Walking (every 255 steps)on overworld in Isle of Armor/Crown Tundra+1+1+1 (up to 180)
Pomeg Berry
Kelpsy Berry
Qualot Berry
Hondew Berry
Grepa Berry
Tamato Berry
+10+5+1
HP Up
Protein
Iron
Carbos
Calcium
Rare Candy
PP Up
Zinc
PP Max
+4+20
Guard Spec.
Dire Hit
X Attack
X Defense
X Speed
X Accuracy
X Sp. Atk
X Sp. Def
+1+10
Health Feather
Muscle Feather
Resist Feather
Genius Feather
Clever Feather
Swift Feather
+2+10
Exp. Candy XS
Exp. CandyS
+1+10
Exp. Candy M
+2+10
Exp. Candy L
+3+20
Exp. Candy XL
Dynamax Candy
+4+20
Sweet Heart
+3+30
Energy Powder
Heal Powder
-5-5-10
Energy Root
Revival Herb
-5-5-10

Brilliant Diamond & Shining Pearl

Level Up+1+20
Faint from Pokémon less than 30 levels higher-1-1-1
Faint from Pokémon more than 30 levels higher-5-5-10
Battle Gym Leader or Champion+5+30
Walking (50% chance every 128 steps)+1+1+1
Walking (50% chance every 128 steps) Pokémon following+1+1+1
Eat a Poffin+1+1+1
Pomeg Berry
Kelpsy Berry
Qualot Berry
Hondew Berry
Grepa Berry
Tamato Berry
+10+5+1
HP Up
Protein
Iron
Carbos
Calcium
Rare Candy
PP Up
Zinc
PP Max
+4+20
Guard Spec.
Dire Hit
X Attack
X Defense
X Speed
X Accuracy
X Sp. Atk
X Sp. Def
+1+10
Energy Powder
Heal Powder
-5-5-10
Energy Root
Revival Herb
-5-5-10

Legends: Arceus

ActionCurrent Friendship
0-99100-159160+
Level Up+3+2+1
Faint-1-2-3
Gain Experience
Break landmarks, win battles, Capture Pokémon
+1+1+1
Razz Berry
Nanab Berry
Pinap Berry
Aux Powerguard
Dire Hit
Choice Dumpling
Twice-Spiced Radish
Swap Snack
Hopo Berry
Aux Power
Aux Guard
Aux Evasion
+1+1+1
Potion
Full Restore
Max Potion
Hyper Potion
Super Potion
Full Heal
Revive
Max Revive
Max Ether
Max Elixir
Old Gateau
Exp. Candy S
Jubilife Muffin
+1+10
Exp. Candy M
+2+10
Exp. Candy L
+3+20
Exp. Candy XL
Rare Candy
+4+20
Remedy
Fine Remedey
Superb Remedy
-15-20-25

Items in Join Avenue, the Juice Shoppe & Festival Plaza will also increase happiness various levels. Putting your Pokémon into Isle Avue in the Poké Pelago in Sun & Moon will also boost happiness for the first 24 hours, before lowering it.

Note: Trading or transferring the Pokémon will reset the Happiness to its Base Value. Each Pokémon has a different base happiness.

And of course, there are things that will frustrate a Pokémon, causing it to dislike you. These actions include:
-Feeding it bitter food
-Letting it faint continuously in battle

If you capture your Pokémon in a Luxury Ball, the increase in happiness is much larger than if in a normal ball.

There's also one item that influences happiness; the Soothe Bell. As for what it does, it increases the amount of happiness gained.

So you're probably wondering (or already know) how to check on your Pokémon's happiness without a fancy equation or whatever. Well, you can easily check on it at one of the houses in Verdanturf Town (Ruby/Sapphire/Emerald/OmegaRuby/AlphaSapphire), Daisy in Pallet Town (FireRed/LeafGreen) or a girl in Goldenrod City. There's a lady in one of the houses in that town that will enlighten you on how much your Pokémon likes you. In Unova, there are two people who will give you a Happiness check. In Nacrene City is one who will give you a broad spectrum while in Icirrus City's Pokémon Fan Club is a woman who will give you the details. In Black 2 & White 2, Bianca will give you details too via the Xtransceiver. In X & Y, it's given by a girl in Laverre City and in Sun & Moon you can find the happiness checker in the road of Konikoni City. The statements are as followed

Statement Values
SM: My! It feels incredibly close to you! Nothing makes it happier than being with you!
ORAS: It’s extremely friendly toward you. It couldn’t possibly love you more. It’s a pleasure to see!
X & Y: It's amazingly friendly toward you! It must be so happy spending every day with you!
B2W2 Bianca: What great friends you are! It's so nice how you trust each other! It makes me kinda jealous!
BW/B2W2: It is super friendly to you! I'm a bit jealous!
DPHGSS: It looks really happy! It must love you a lot
Plat: It simply adores you! Why, I feel like I'm intruding!
FRLG: It couldn't possibly love you any more than it does now. Your Pokemon is happy beyond words.
RSE: It adores you. It can't possibly love you any more. I even feel happy seeing it.
250+
SM: You clearly love your Pokémon, and you must spend a lot of time together.
ORAS: It seems to be very happy. It’s obviously friendly toward you.
X & Y: You must really like your and always keep it by your side!
B2W2 Bianca: You two sure get along great! It looks like you are having fun being together! You seem bright and cheerful!
BW/B2W2: It is quite friendly to you! You must be a kind person!
DPHGSS: I get the feeling that it really trusts you
Plat: It's very friendly toward you. I can tell you treat it kindly.
FRLG: It looks very happy.
RSE: It seems to be very happy. It obviously likes you a whole lot.
200-249
SM: Hm. I’d say that you and Pokémon have the potential to be an even greater combo.
ORAS: It’s quite friendly toward you. It seems to want to be babied a little.
X & Y: Well, I think you and ____ will be an even greater combo someday!
B2W2 Bianca: You seem really close! You look so, so happy! It's enough to make me happy too!"
BW/B2W2: It is friendly to you. It must be happy with you.
DPHGSS: It's friendly toward you. It looks sort of happy
Plat: It's quite friendly to you. It must be happy being with you.
FRLG: It's quite friendly with you. Keep being good to it!
RSE: It likes you quite a lot. It seems to want to be babied a little.
150-199
SM: Hmmm. I think...it feels friendly toward you. At least a little...
ORAS: It’s getting used to you. It seems to believe in you.
X & Y: It's a little bit Friendly to you... Something like that.
B2W2 Bianca: You still have room for improvement. How nice! I mean, you can become even better friends!.
BW/B2W2: It is a little friendly to you... That's what I'm getting.
DPHGSS: It's quite cute
Plat: It's warming up to you. That's my impression.
FRLG: It's warming up to you. Trust must be growing between you.
RSE: It's getting used to you. It seems to believe in you.
100-149
SM: Hmmm... I’d say that you and your Pokémon still have a long way to go.
ORAS: It’s not very used to you yet. It neither loves nor hates you.
X & Y: Hmm...I think you have a lot of time ahead of you to get to know one another better
B2W2 Bianca: You're starting to get to be friends. Just maybe, walking along the same path has made you understand one another
BW/B2W2: The relationship is neither good nor bad... It looks neutral.
DPHGSS: You should treat it better. It's not used to you.
Plat: It's feeling neutral toward you. It's up to you to change that.
FRLG: It's not quite familiar with you yet.
RSE: It's not very used to you yet. It neither loves nor hates you.
50-99
SM: What is going on here? Do you let it get knocked out a lot in battles or something? This is bad.
ORAS: It’s very wary. It has a scary look in its eyes. It doesn’t like you much at all.
X & Y: Don't tell me! Are you just letting it get knocked out in Pokémon battles?!B2W2 Bianca: You aren't getting along, are you? It's glaring at you with a look that's kinda scary
BW/B2W2: Hmmm... It may not like you very much.
By any chance, you... Are you a very strict person? I feel that it really doesn't like you...
DPHGSS: It doesn't seem to like you at all. It looks mean
Plat: Hmm... It seems to dislike you a little.
FRLG: I don't like the way it glares at you. Could you try being a little nicer to it?
RSE: It's very wary. It has scary viciousness in its eyes. It doesn't like you much at all.
1-49
SM: Oh dear. You must be a merciless Trainer... Do you use Frustration or just not know better?
ORAS: This is a little hard for me to say... Your Pokémon simply detests you. Doesn’t that make you uncomfortable?
X & Y: Don't tell me! Are you just letting it get knocked out in Pokémon battles?!B2W2 Bianca: You aren't getting along, are you? It doesn't look like it's having fun even when you're together
BW/B2W2: Hmmm... It may not like you very much.
By any chance, you... Are you a very strict person? I feel that it really doesn't like you...
DPHGSS: It doesn't seem to like you at all. It looks mean
Plat: Hmm... It seems to dislike you a little.
FRLG: I don't like the way it glares at you. Could you try being a little nicer to it?
RSE: It's very wary. It has scary viciousness in its eyes. It doesn't like you much at all.
0

In Diamond, Pearl & Platinum, there is a Pokétch application that will show you the happiness of your Pokémon in your party simply by tapping them. It will be shown as hearts. There are various levels including one small heart, two small hearts, two big hearts and two large hearts. The two large hearts being the optimum happiness level. There is also a person, Dr. Footstep in Route 213, who'll give you the footstep ribbon if you show him a Pokémon with maximum happiness

So as for Return/Frustration, they work best when they're happiest/unhappiest respectively. Now seeing as it's so simple to gain happiness, I don't see why you'd want to use Frustration at all. Anyway, they're both Normal attacks and when the happiness is maxed out, the base attack is 104.

From Pokémon Let's Go, Pikachu & Let's Go, Eeve onwards, there are also special bonuses that can happen if friendship is at a certain value. These were tied to affection prior to this but combined with friendship

Special Bonuses

Happiness Value Bonus
255 10% Chance of Evading Attack
255 2* Critical Hit Rate
255 20% Chance of surviving a KO
220-255 20% of healing status
220-254 15% Chance of surviving a KO
180-219 10% Chance of surviving a KO

To evolve Pokémon via happiness, you must reach 220 it by doing the above efforts, once that is done, you can evolve your Pokémon by making it raise a level either through battle or a Rare Candy. Below are all the Pokémon that evolve through Happiness and what they evolve into:

Lower Stage Pokémon Name Pokémon Name Final Stage
#042 Golbat #169 Crobat
#052 Alolan Meowth #053 Alolan Persian
#113 Chansey #242 Blissey
#133 Eevee #196 Espeon
Only in Daytime
#133 Eevee #197 Umbreon
Only in Night
#133 Eevee #700 Sylveon
Only with a Fairy Move
#172 Pichu #025 Pikachu
#173 Cleffa #035 Clefairy
#174 Igglybuff #039 Jigglypuff
#175 Togepi #176 Togetic
#298 Azurill #183 Marill
#406 Budew #315 Roselia
Only in Daytime
#427 Buneary #428 Lopunny
#433 Chingling #358 Chimecho
Only in Night
#446 Munchlax #143 Snorlax
#447 Riolu #448 Lucario
Only in Daytime
#527 Woobat #528 Swoobat
#541 Swadloon #542 Leavanny
#772 Type: Null #773 Silvally
#872 Snom #873 Frosmoth
Only in Night

And thats all there is to this relatively simple game mechanic.

Thanks to Dragonair for writing this for us