My expertise is mainly with English food, but here's some info. Oh, also, if you like food history try /r/AskFoodHistorians/

Coffee comes to Europe through trade with the Ottoman Turks. Coffee first came from Ethiopia, sometime in the 15th century it was introduced to Yemen (The Ottomans and the Yemeni Coffee Trade, Hathaway, Jane, 163) and by the 16th century was also being grown in Egypt, Syria, and near Istanbul (Ibid.). If those places sound like they might be linked at the time it's because they were all part of the Ottoman Empire. They loved their coffee, and the best coffee was grown in Yemen (164).

Now getting to your question about Europe.

Though I've heard anecdotally that coffee first came to Europe in Malta in the 1560s the earliest I've been able to find information on is in 1573 (All About Coffee Ukers, William Harrison, 2). Unfortunately actual introduction to Europe isn't until 1615 (25). However Europeans who visited the Ottoman Empire had been drinking it in coffee shops as early as 1582 (ibid.). The Venetians, Prospero Alpini to be specific, are the ones to first talk about the method of preparation for coffee in 1592 (26). By 1615 the Turkish drink was well known in Europe though, and was finally being sold in Venice, though mainly for medicinal purposes (27). By 1645 Italian drink vendors were selling coffee and drinking chocolate as well as lemonade and alcohol (ibid.) and by 1671 coffee was a prominent drink. Although there are rumors of earlier ones the earliest confirmed coffee house, in that they sold primarily coffee, is in Venice in 1683.

Now moving to England, in 1600 William Biddulph was the first Englishman to write about coffee. He wrote about how the Turks/Ottoman Empire loved it, similar to the early Italian information (Coffeehouse, Cowan, Brian, 5). By 1650 Coffee had come to England as well, being sold similarly to in Italy five years earlier (6). Coffee houses came later, about the early 17th century, though in England they also sold chocolate and tea (81) as well as tobacco (83).

I don't know how coffee was drunk in Italy, though presumably it would be more similar to how the Turks drank it. In England, however, the coffee was a bit weaker than modern coffee. They used a ratio of between 1 ounce coffee per quart to 3 ounces per quart depending on the coffee house (80). The coffee was unfiltered though with the grounds mixed right into the boiling water and left to steep for a bit with sugar and frequently milk (ibid.). If you want to try it get out your French Press and make a pot with the ratio above, but don't use the filter/plunger. You'll get a reasonable approximation when you pour it out. The coffee will have settled a bit, but there will still be grounds in your cup.