I was looking at a map of Imperial Rome http://en.wikipedia.org/wiki/File:RomanEmpire_117.svg (Wikipedia, I know)
And it shows that Asia was just the western province in modern-day Turkey. How did Asia come to mean the entire east of Europe?
"Asia" is from a Greek word, refering to the land to the east:
speculated to be from Akkadian asu "to go out, to rise," in reference to the sun, thus "the land of the sunrise."
Here are some previous posts discussing the naming of continents
When were the continents first given names, what were those names and who gave them those names?
Why are Europe and Asia considered separate continents, and was there ever a time when they weren't?