When was the earliest point in time we had a fairly accurate (to today's standard) map of the world?
Ptolemy's map was one of the first maps of the world. Ptolemy's Geography (written 1st century AD) was a description of the world at the time; this was a text, not a map. There were attempts to translate Ptolemy's Geography into a map in the 15th century, and the result is the image above.
There was a significant emphasis on cartography during this period for a few reasons. Firstly, rulers who had empires wanted to know more about their own empires; it would be helpful for them to visualize their empires through a map. Also, diplomacy developed through trade; one needed maps to trade effectively, and rulers were interested in knowing the empires around them. Further, it would be easier for your ambassador to get to another country and begin forming diplomatic relations and alliances if they had a map.
Source: "Early Modern Europe 1450-1789" by Merry Weisner-Hanks