Countries have been issuing postage stamps for over 175 years. The classic period of stamp issues are from 1840-1900. And the modern period is from 1960 until now. The Golden Age of philately encompasses the issues from 1900-1960. These are, for most countries, the most popular issues and usually the most interesting as well.
By 1900, philately was already a well established hobby with tens of thousands of collectors worldwide. Postal agencies saw stamp collectors as a revenue source and began competing for their collecting dollar by issuing attractive limited issue sets to tempt collectors into buying them for their collection. Most of the most popular British, French, Portuguese, Spanish, and German colonial sets are from this period. The United States issued many of its best designed and most popular stamps. Fifteen or more Latin American countries issued tens of thousands of stamps and intentional and unintentional varieties. Japan issued many of its most sought after stamps, and of course other Asian issues have become a mainstay of the best sets of the hobby.
WWI and WWII contributed to the philatelic interest of this period. The scores of new issuing entities of Europe that the war produced (such as Saar, Danzig, and more) are fiercely popular and interesting collecting specialties. The war in Asia and Japanese Occupations along with the Chinese Communist Revolution make for fantastic philately. And the liberation of scores of British, French, and Portuguese colonies after WWII means that this Golden Age has the first independence issues of such countries as India, Kuwait, Togo, and Burma. Russia, too, has some of the world’s most interesting issues in this period, from the Revolution issues to the wonderfully scarce and popular Soviet period sets from 1930-1960. The world’s first Airmails were issued in the Golden Age, and they by themselves could be a lifelong philatelic endeavor. All in all, if a collector had only one period to collect, the Golden Age would be the best choice of all.