Golden Ages are periods of great flourishing for an empire, periods during which everything seems to be working just perfectly as if the gods have suddenly smiled upon us.
In Civilization V, Golden Ages are periods during which your Civilization receives bonus Production, Gold, and Culture for a limited number of turns. 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 over a period of many turns. The excess Happiness (i.e. every count of 1 above the 0 point) will be added directly into a Golden Age counter every turn. Note that if your empire is Unhappy, your Golden Age counter doesn't diminish, so don't worry about having to start your Golden Age quota from scratch once you manage to get your empire's Happiness back up.
Once the required amount of Happiness has been accumulated, a Golden Age will dawn. During this time, all land which produces at least 1 Gold when worked will produce 1 more, and Production and Culture generation are increased by 20% in all cities. During a Golden Age, your excess happiness will not count toward the next Golden Age; the counter 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. Also, the number of cities in your empire increases the Golden Age quota number - the more you have, the more Happiness you'll need for the next Golden Age. The increase is not significant, though.
Golden Ages normally last for 10 turns, but there are also gameplay effects that will increase the duration of Golden Ages. The unique ability of the Persian civilization increases the length of Golden Ages by 50% and provides 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 this period, extra happiness again won't contribute to your normal counter, but in this case, after the end of the Golden Age, the counter resumes from the point where it was interrupted, not 0. In other words, there is always one and only one Golden Age counter, which is related to the "normal" Golden Age, and is not affected by "bonus" Golden Ages. Note however, that the "bonus" Ages still increase the amount required for the next "normal" Age.
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 Gods & Kings, this ability becomes the sole purview of the Great Artist. His Golden Age lasts for 8 turns and doesn't diminish in length for successive uses, no matter how many Golden Ages you start with Great Artists. Again, he is consumed in the process.
Golden Ages are an important period of boost in your empire's most important activities - use them to gather more Gold in your treasury by assigning Citizens to work all gold-producing tiles, and to complete lengthy or important projects faster.