Theory of Happiness

Happiness is a measurement of how much a Pokémon loves its Trainer and enjoys being in their care.
As of Diamond and Pearl, Happiness is inheret in all Pokémon as a value ranging between 0 and 255.
All wild Pokémon have the same starting Happiness value of about 70 (or 27%). Pokémon that are hatched from eggs get a default Happiness value of 120 (47%). If a Pokémon is caught in a Friend Ball, Happiness is set to 200 (78%).
After the concept of Pokémon evolving on Happiness was introduced, Happiness-based evolutions occur when Happiness reaches 220 (86%).

Now, raising Happiness can be done in any number of ways, including-

Gaining a level (+2)
Walking 256 steps (+1)
Stat modifiers (+2)
Using Soothe Bell (+1 for each other positive event)
Massage in Veilstone City by girl in house nearest to Veilstone Gym (+3)
Massage at Ribbon Syndicate in Resort Area
Using Potions and healing items

However, there are items that will make your Pokémon's Happiness level go down-

Energypowder (-5)
Heal powder (-5)
Energy root (-10)
Revival Herb (-15)
Faint (-1)
Trading a Pokémon (reset to 70)

There are several ways to tell if your Pokémon is very attached to you.

In Eterna City, a girl in the Pokémon Center will give away a Pokétch application that shows how much the Pokémon in your party likes you. Maximum happiness is represented by two big hearts.

Dr. Footstep, the sole resident of the Footstep House on Route 213, will also tell a Pokémon's happiness, albeit in a roundabout way which differs among Pokémon species.

Moves affected by Happiness
There are also moves where the base power are affect by Happiness:

Return- is a Normal-type move that deals a varying amount of damage. The damage inflicted depends on the user's happiness stat, but when maxed out (255), its base power is 102.
Frustration- is a Normal-type move that deals a varying amount of damage, according to the following formula:
Power = (255 - happiness) / 2.5
At zero (0) happiness, this attacks power is also 102.

Happiness Evolution
There are a few Pokémon that evolve when their Happiness value is at or above 220 and they are leveled up:

Golbat → Crobat
Chansey → Blissey
Eevee → Espeon (during the day)
Eevee → Umbreon (during the night)
Pichu → Pikachu
Cleffa → Clefairy
Igglybuff → Jigglypuff
Togepi → Togetic
Azurill → Marill
Budew → Roselia (during the day)
Buneary → Lopunny
Chingling → Chimecho (during the night)
Munchlax → Snorlax
Riolu → Lucario (during the day)