Organizing the pious feelings of the masses with the effective leadership of a central church has many benefits. Most of these, however, are evident in the two main religious buildings, which reflect the new level of dedication of your society in every detail: architecture, decoration, splendor, wealth. This, in turn, increases their own Faith effect over the populace.
Being the requirement for mostly all other Policies in the tree, adopting Organized Religion first is usually the sensible choice. Besides unlocking higher-end Policies, the additional Faith per building is great in the beginning of the game - both for founding a Pantheon and later, a Religion. Note, however, that if you have no religious buildings at all, you will be wasting a Policy slot, so aim at building at least one Shrine before - or right after - adopting this Policy.
Organized religion is one in which a number of people follow the same religious practices and have the same beliefs. The religion may have priests who intercede with higher powers on behalf of the believers or who set/interpret the religion's rules. There may be a complicated hierarchy of priests and other religious leaders. The religion may have temples or other sanctums in which the people pray and worship. Generally the believers contribute money to support the religion, its priests and buildings.
Generally, organized religion is a force of stability within a culture. People who share a religion have something important in common, and although strangers are less likely to murder each other. It can also be quite divisive as well - historically many organized religions have seen other religions as threats or abominations, and more than one war has been fought over whose god is God.