There's a general, vague feeling that the Renaissance came after the Middle Ages. But this is not the case. The Renaissance is actually just a name for late medieval Italy.
In addition, in spite of a sense that it must have marked something new and improved (after all, we use the word Renaissance, meaning rebirth), the Renaissance began with the Black Death of 1347 and continued with economic disaster. The art that we now see as the most glorious achievement of Renaissance Italy (and it is) really appears only a century or so after self-styled Italian humanists declared they were living in a new time, a Rinascimento (Italian for Renaissance), and should be seen as a defiant cry against the despair of the age.
As I discussed in an earlier post on medieval Italy, the peninsula was the most urbanized part of medieval Europe, a major trade center where luxury goods from the East first reached Europe. This focus on trade meant that intellectual life took something of a back seat--why think deep thoughts when one could be making money? Italy did have Europe's first university, Bologna, but it concentrated on law, not philosophy or the classics--these were studied instead at northern universities, especially Paris (see more here on medieval universities).
So when the trade routes collapsed in the aftermath of the plague, the always energetic Italians looked for something new to concentrate on and focused on ancient philosophy and the arts. They were, not surprisingly, especially enamored with ancient Rome. Scholars set out to try to discover "pure," classical Latin, without any of the words medieval people had added to it in the last thousand years. They started building their churches and monuments in a neo-Roman style. At parties they would dress Roman and address each other as Cicero or Horace (a little different from the modern vision of toga parties…)
These scholars called themselves humanists, umanistici. Note that although in modern usage we often use the term "humanist" to mean a secularist, as opposed to a religious person, the Italian humanists were intensely religious. For them, the term meant someone interested in ancient philosophy and ideas.
One sometimes sees the term "northern Renaissance." This is specifically an art-history term, meaning the flourishing of art in places like Flanders in the sixteenth century, or the new interest in reading Greek and Latin works in the original, spurred by the invention of the printing press. This "northern Renaissance" overlaps with the Protestant Reformation, which started in the first decades of the sixteenth century and led to such disasters as the Thirty Years War (unfortunately, even now, religion has always seemed like a good thing to have a war over.)
For a lot of people, life was worse in the Italian Renaissance than it was earlier. With the new interest in Rome, slavery was revived, after not being seen in western Europe for centuries, because Rome, after all, had been a slave society. In imitation of patriarchal Rome, Renaissance husbands tried (without notable success) to make women obey. The cities of the Renaissance, in a time of turmoil and huge discrepancies of wealth, were violent, dangerous places.
It is interesting that the belief that the Renaissance came after the Middle Ages is so strong that some textbooks have a chapter on the fifteenth century--including such things as Joan of Arc, the end of the Hundred Years War, and the War of the Roses in England--before rather than after the chapter on the fourteenth-century Renaissance.
The Italian humanists, convinced they represented something new (they indeed invented the term Middle Ages, dividing history into antiquity, their own glorious modern period, and the age "in the middle"), would have been shocked to discover that events around the year 1500, including the printing press, Columbus's discovery of the New World, and the Reformation marked a bigger change than did their toga parties.