Golden Ages are periods during which your Civilization when your empire produces extra Production, Gold and Culture. In addition, there will be no periods of Anarchy if you decide to adopt a social policy which is mutually exclusive with an existing policy.
The normal process of entering a Golden Age consists of generating more Happiness than Unhappiness over a period of many turns. The excess Happiness will feed directly into a Golden Age quota. Once the required amount of Happiness has been accumulated, a Golden Age will dawn. During this time, all tiles which produce at least 1 Gold when worked will produce 1 extra (for example, if a tile produces 3 gold, it will provide 4; if 1, then instead 2) and Production and Culture generation is increased by 25% in all cities. During a Golden Age, your excess Happiness will not count toward the next Golden Age; the quota will begin again at zero once the Golden Age ends. Every subsequent Golden Age will require more and more Happiness to be accumulated than the Golden Ages which preceded it.
Golden Ages will normally last for 10 turns. Certain Wonders will increase the duration of Golden Ages. The unique ability of the Persian civilization will also increase the length of Golden Ages by 50% and provide additional military bonuses for the entire duration. All of these percentage bonuses stack and are applied all at once to the base duration of 10 turns; Golden Ages can potentially last for 20 turns or more with the appropriate bonuses.
Certain Wonders and social policies can also begin a Golden Age. During it, extra happiness again won't contribute to your normal quota, but in this case, after the end of the Golden Age, your normal quota resumes from the point where it was interrupted, not 0.
Previously, in Civilization V, most Great People could initiate a Golden Age for a shorter time than the normal one. The Great Person was consumed, and what's more, every time another one was used to start a Golden Age, its duration would diminished even more with each successive time this was done, down to a minimum of three turns. In this ability is now the sole purview of the Great Artist. His Golden Age goes for 8 turns which don't diminish for successive uses, no matter how many Golden Ages you start with Great Artists. Again, he is consumed in the process.