Happiness in Forge of Empires directly affects how many supplies and coins your citizens produce. Each citizen demands one happiness. If you provide one happiness per citizen, your productivity will be 100%. Drop below that ratio and your citizens will become angry, which reduces output to 50% of the normal amount. When you provide at least 1.4 happiness per citizen, your population will become enthusiastic, which increases productivity to 120%. An enthusiastic population is ideal, although at very early stages in the game it can be difficult to attain or maintain.
In Bronze Age, decorations like memorials (2x2) and cultural buildings such as the stone circle (3x3) are the first options for gaining happiness. The footprint needed for these can quickly add up to the point where you may need to decide between enthusiasm or a few goods or military buildings. Depending on how many coins and supplies you have, you may choose to forego enthusiasm for a time to increase the space you have available for other aspects of the game.
No matter what happiness percentage you find to be ideal, eventually most towns will need additional happiness. This can be accomplished through the following methods:
Decorations
Roads
Great Buildings
Oracle of Delphi
Hagia Sophia
Notre Dame
Frauenkirche of Dresden
Alcatraz, Space Needle
Atomium
Lotus Temple
Gaea Statue
Cultural Buildings
Event Buildings
In general, it is best to prioritize happiness-producing buildings in this order: Great buildings, event buildings, cultural buildings, and finally decorations. Great buildings and event buildings typically provide at least one other benefit besides happiness, making them more efficient than cultural buildings which tend to be large and provide only happiness. Whenever possible, avoid decorations. Decorations take up less space individually, but they are less efficient per tile than other options and do not provide any additional benefits. Most importantly, however, is that decorations will use the valuable polishing and motivating that your other buildings could benefit from instead.
If you do find yourself building decorations and cultural buildings to sustain your population's happiness for a time, try to build larger, rather than smaller, yet still efficient buildings. This will reduce the number of polishes needed for your decorations overall. For instance, in Bronze Age, one memorial (2x2) provides 72 happiness, which is 18 happiness per square (72 / 4 = 18). One tree or obelisk also provide the exact same amount of happiness per square, but four trees requires four polishes, while a memorial only requires one. Having motivated residential and production buildings (instead of decorations) early in the game will be beneficial, because motivated buildings have double the output of non-motivated buildings.