Europe in 1300
Islam
The principal powers in the Islamic world that centered on the Mediterranean were: the Almoravids in Spain, the Mamluks in Egypt and in Syria, and a new power, about to come onto the scene in 1300, the Turks. I've already spoken of the Almoravids, so I'll talk about the other two here.
The Mamluks
The Mamluks began as slaves of the Ayyubids of Egypt. They were non-Muslims captured as children, converted to Islam and serving the rulers as soldiers. In the 13th the slaves overthrew the masters and the Mamluks became the ruling dynasty all through our period (they were overthrown by the Ottomans in 1517; ironically, many of the original Mamluks were in fact Turkish). The word denotes the condition, not a people, so it's possible to read about mamluks (spelling varies) as far away as India and as late as the 1800s. The ones with which we are concerned are those that comprised the Sultanate of Cairo.
Even after the rise of Baybars, the first Mamluk sultan, the Mamluks continued to be slaves, an exclusive military faction within Egyptian society. These were the Mamluks who defeated the Mongols in 1260, drove the Christians out of the Holy Land in 1291, and who defeated the Mongols repeatedly until reaching a peace agreement with them in 1323.
The Mamluks continued to rule Egypt, and sometimes Syria. They came into conflict with the Ottoman Turks in the 15th century, though, and that century saw a long and steady decline in Egypt.
Ottoman Turks
If there is any dynasty that best fits our arbitrary dates of 1300 to 1500, oddly enough it would be a non-European, non-Christian people, the Ottomans. Their history begins with Osman I, who established his dynasty right around 1300, in western Turkey. He was able to do this mainly because of the weakness of the Byzantine Empire, but also because the Ottomans were exceptionally well organized and were fierce fighters.
Ottoman expansion to the east and south was bounded by Mamluk and Mongol power, and was disrupted for a time by Tamerlane (around 1400). To the north, though, there were opportunities in the decaying Byzantine Empire. Constantinople itself was a tough nut to crack, so the Ottomans, after securing Anatolia, expanded into Greece and the Balkans, and that is the primary story of that region right through the 14th century. Increasingly during our course we will find Europeans, and especially the German Emperors, being preoccupied with the "Turkish menace".
It is not uncommon, moreover, for historians to give at least some credit to the Turks for closing off trade routes to the East and thereby causing Europeans to seek other ways of getting to the eastern markets. This used to be a very explicit cause and effect: the Turks took over the Near East, closed trade routes, causing the Portuguese and Spanish to explore the Atlantic. More recently, historians have painted a more nuanced picture that includes desire for gold in west Africa and a strong impulse to convert the heathen. Nevertheless, the economic factor was certainly significant.