The Black Death started in 1346 when rats from China started to flee the dried-out grasslands there. Previously, there had been a serious famine in the region, and the lack of rain dried up the grasslands and forced the rats to move.
Now, the rats were not the problem, but the fleas living on the rats were. These fleas were infected with a bacteria called Yersinia pestis.
During this time, some Genoese merchants were trapped inside the Genoese colony called Caffa (today called Feodosiia in Crimea). These Genoese merchants angered the local Arab population, who called upon the Mongols to punish the traders.
During a siege of the city, the Mongols were infected by the disease and started to die. Seeing that they could not prolong the siege any longer, they catapulted the infected soldiers' dead bodies over the wall of the city.
When the plague started killing people inside Caffa, the merchants decided it was high time to leave, so they boarded their ships and sailed away. Unluckily, more infected rats got on the ship and came with the merchants until they reached Sicily, where the rats disembarked and began spreading the plague.
From Sicily, the plague spread to Spain, France, England, and many other populous nations. It spread to Germany, Scotland, and Scandinavia, even reaching Iceland and Norway.
The plague spread quickly because of the living conditions, trade routes, and general lack of hygiene in the Middle Ages. Countries with less-developed trading, like the Basque Country and Belgium, were not affected as much, and the Netherlands was mostly plague-free, due to many swamps drying up and leaving the inhabitants plenty of space to live.
The Black Death was the most devastating plague in history, having killed around 200 million people, or around 50% of the world's population at the time.