For every unique liked item that is placed two squares or less adjacent to the monster, happiness is increased by 25%. (Two or more of the same item/monster only increases happiness once.) Thus four items are required to make a monster 100% happy. Note that on some islands, some monsters have five likes available, but it is not possible to make a monster more than 100% happy.
Happiness affects the rate of coin production. For each increase of 25% happiness, the coin production rate increases by 25% of the base value. Thus, if a monster is at 100% happiness, it will produce coins at twice the rate of a monster at 0% happiness. If the monster is at least 25% happy, the "rate" value on the info screen will display in green numbers. Happiness does not affect a monster's maximum income, i.e. the number of coins the monster will hold.
As a matter of strategy, it is worth considering how often you collect coins from your monsters. For example, a level-15 Potbelly reaches its coin limit in 6 minutes if it is 0% happy, and in 3 minutes if it is 100% happy. If you never collect coins from it more frequently than every 6 minutes, you get no benefit from increasing its happiness. Most other monsters take much longer to reach their coin limits, so improving their happiness may increase the number of coins you can collect. The wiki page for each monster shows its coin limit for each level, as well as its earning rate for each % Happiness. If you divide the coin limit by the earning rate, you get the time needed for the monster to reach its coin limit. Or, if you prefer, the wiki page also shows the times for the monster at level 15 to reach its coin limit for 0% Happiness and for 100% Happiness, and those times will be pretty similar for the monster at any level.
For purposes of making other monsters happy, a Rare monster is the same as its normal version. For example, a Toe Jammer likes a Furcorn, and also likes a Rare Furcorn. If a monster is close to both the normal and Rare versions of a monster it likes, its Happiness increases only by 25%, not by 50%.
If an item is placed near a monster that makes it more happy, an upwards harp-strum noise will be heard, and pink hearts will fly out of the monster's head. Conversely, if an item is removed and the monster becomes less happy, a downwards harp-strum noise will be heard, and blue/black broken hearts will fly out of the monster's head.
The Unity Tree nullifies the requirement to place an item within two squares of a monster to make it happy. Items and monsters can then be placed anywhere on the island and still increase happiness. This allows for greater freedom in designing islands. However, if an item a monster likes is sold, the monster gets sad.
If a liked object is put away inside a Storage Structure, it will no longer help to make the monster happy. The same applies if the Hotel Structure is used to store a liked monster. If you have a Unity Tree, these limitations go away.