The Middle Ages is the term applied by Renaissance writers to the time between the glorious ancient world and their own brilliance. Dates vary simply because the Middle Ages happened at different times in different parts of Europe. The term medieval may apply to anything from the Fall of Rome, which itself is a messy date, to the start of the Renaissance. Italy is a good place to mark as the centre of Europe during this time, because Europe became medieval as the Roman Empire fell back to Rome, and the Renaissance began in the city states of northern Italy and spread outwards. Thus England, being far from Italy, was medieval for much longer than Florence or Venice.
To generalise grossly but give an overview, the early part of the Middle Ages, the Dark Age, is marked by the collapse of an Empire, invasions from all directions and small, warring territories. Tribal, clan-based ruling was usual, and warrior princes struggled to carve out kingdoms to rule. The Christian Church had little power in most places, and the literacy and science of the ancient world was largely forgotten.
After a few centuries of this, Charlemagne became the Emperor of a huge country in what is now part of France and Germany. After his death the kingdom was divided between his three sons, and both France and the Holy Roman Empire claimed descendance from Charlemagne's realm.
This roughly marks the end of the Dark Ages. Gradually, various small states became dominant and the map of Europe began to take on some sort of coherence from generation to generation. The feudal system developed, more in some places than others, and when Duke William of Normandy invaded England in 1066, Christianity had regained it's hold on Europe such that not only did William wait for the Pope's blessing to invade, but he had his soldiers do penance for killing before the battle of Hastings.
The High Middle Ages followed in the western end of Western Europe. Gothic architecture, complex feudal laws and the flowering of the Chivalric Ideal ran parallel to overpopulation, church corruption and the Black Death.
Meanwhile, over in Italy, the city states had already entered the Renaissance when the Black Death struck in the late 1340s. Science, art, politics, you name it, they were leading it.
Over the next century, the west of the West came around to the idea of Renaissance. In England the end of the Middle Ages is often taken to be 1485, the year when the last medieval, Plantagenet monarch was defeated by the first Renaissance monarch, Henry VII. This is a convenient date, much better than 1492 or 1517 because it marks a change in attitude. For Columbus to discover America or Luther to agitate for church reform, the desire to fund exploration or dissatisfaction with the church already had to be there. Columbus' voyage was funded by a Renaissance attitude, one that had already sailed around Africa and wanted more.
Like all historical generalisations, this is only true to a limited extent. Every country had a different experience during this time, and to take this brief summary seriously would be folly. The Middle Ages were a complex, fascinating time, much like our own in many ways. For further reading, consider the Crusades, the Plantagenet dynasty, feudalism. Remember, too, that the Middle Ages only applies to a small part of the world. The Eastern Roman Empire, that is, Constantinople, was much more advanced than the West, and retained much of the glory of the ancient world while progressing to the Renaissance. The spread of Islam was also important in this time - most of the Iberian Peninsula was Muslim at some point - and as usual, Western Europe was the most backward and ignorant civilisation on Earth, even if it did have nice cathedrals and King Arthur.