Every civilization can experience a Golden Age, a 20-turn period in which every citizen laborer in the empire is more productive, producing one extra shield and/or commerce if already producing at least one. There can only be one golden age per civilization, even if you meet the requirements for it multiple times. A civilization's Golden Age is triggered in one of two ways:
Unique Unit Victory
Each civilization has a unique military unit -- one that only it can build. A civilization's Golden Age can begin when its unique military unit is victorious in battle.
Each Great Wonder is associated with one or more civilization strengths. If a civilization builds a Great Wonder (or Wonders) that corresponds to both its unique Strengths, its Golden Age can be triggered.
For example, if the Egyptians (industrious and religious) build the Pyramids (which is both a feat of construction and a place of worship), Egypt's Golden Age could begin. Or if the Babylonians (scientific and religious) build the Great Library (scientific) and the Oracle (religious) Babylon's Golden Age could begin.
Scientific Golden Age
This feature is broken in all commercial versions of civilization 3 and was never officially patched. Theoretically, a Golden Age of Science is triggered by a Scientific Great Leader that provides a 25% boost to all research output for 20 turns. This boost is reflected in the UI but does not actually affect the research as it's calculated differently. It is therefore a waste of time to use this ablity and should be disregarded.