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 include:
-Having the Pokémon in your party
-Giving it Vitamins

-Giving it a massage in Pallet Town, Goldenrod City, Veilstone City or Ribbon Syndicate
-Giving it Certain Berries -Gaining a Level

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), 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 Shippou City is one who will give you a broad spectrum while in Sekka City's Pokémon Fan Club is a woman wYou know that your Pokémon likes you as much as possible when she says, "It adores you, it cant possible love you anymore. I even feel happy seeing it." The statements are as followed

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

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
	#113 Chansey	#242 Blissey
	#133 Eevee	#196 Espeon
	Only in Daytime
	#133 Eevee	#197 Umbreon
	Only in Night
	#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

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

Thanks to Dragonair for writing this for us