The 17th century.

The widespread use of coffee starts in the middle of the 15th century, when farmers in Yemen began cultivating plants brought from wild sources in Ethiopia. From Yemen, a prominent trading hub in the Islamic world, it spread to Mecca and Cairo in the first decades of the 16th century. By 1555, it was present in Istanbul, and in 1582, the German physician Leonhart Rauwolf wrote the first description of coffee by a European.

Brian Cowan's The Social Life of Coffee: The Emergence of the British Coffeehouse describes the first English-language description: "In a letter from Aleppo, written in 1600, the clergyman William Biddulph became the first Englishman to write about coffee. He noted that the Turks’ 'most common drinke is Coffa, which is a blacke kinde of drinke, made of a kind of Pulse like Pease, called Coava, which being grownd in the Mill, and boiled in water, they drinke it as hot as they can suffer it.' Ten years later, George Sandys put his own observations on this strange Turkish beverage in print; he found it 'blacke as soote, and tasting not much unlike it.'"

In the 17th century, coffee moved through three principal phases: A foreign curiosity, a drink of the elite, and a drink of the everyday man. By the end of the century, thousands of coffeehouses were open in cities across Europe and had become havens for discussion, shelters for playwrights and authors, and the "third place" famously referred to by Ray Oldenburg.

In 1671, Philippe Sylvestre Dufour, a French pharmacist, published a book named De I'usage du cafe, du the, et du chocolate. This described coffee, tea and cacao as they were all used at the time. Each substance had come to Europe in the 17th century in a big way, and the arrival of each was the product of a series of converging forces. The first of these was the globalizing force of colonialism, which sent ships and armies across the world. In the Middle East and Asia, these forces established trade routes since outright conquest was out of the question at the time.

The second force was the growing wealth of the middle and lower classes. If those ships brought home tea, coffee and chocolate, there was a populace that could afford it and could afford regular purchases of addictive caffeine.

The third force was interest in the wider world and foreign cultures. Dufour's book, for example, contains an illustration of a Middle Eastern man consuming coffee; a Chinese man drinking tea and a Native American drinking chocolate. Exoticism was running in high fervor.

It should be noted that writer Sidney Mintz coined the phrase "drug foods" to include caffeine beverages as well as imported sugar-laden foods like chocolate, and tobacco.

I've previously written about the spread of modern coffee culture in Seattle and I'd again recommend Uncommon Grounds (read the first chapter online here) but the development of 17th century coffee culture in Europe has received extensive coverage from a variety of authors including Fernand Braudel. I've relied on a paper called "The Essence of Commodification: Caffeine Dependencies in the Early Modern World" (Journal of Social History, 2001) for parts of this answer.