Golden ages are triggered when either your civ's unique unit wins combat, or when you build wonders with both of your civs traits. Take Egypt, for example. Industrious and Religious. If they built the Pyramids, which is an industrious and religious wonder, they get a golden age. Or, if they built the sistine chapel (religious, right?) and hoover dam (industrious), they would get a golden age.
You can only have 1 golden age per game, and it lasts 20 turns. During that time, each space produces 1 extra shield, food, and commerce (something like that, i believe).