According to Wikipedia, this division was first put forward in the 18th century by Philip Johan von Strahlenberg. It's best if I just quote the passage in full:
In ancient times, the Greeks classified Europe (derived from the mythological Phoenician princess Europa) and Asia (derived from Asia, a woman in Greek mythology) as separate "lands". Where to draw the dividing line between the two regions is still a matter of discussion. Especially whether the Kuma-Manych Depression or the Caucasus Mountains form the southeast boundary is disputed, since Mount Elbrus would be part of Europe in the latter case, making it (and not Mont Blanc) Europe's highest mountain. Most accepted is probably the boundary as defined by Philip Johan von Strahlenberg in the 18th century. He defined the dividing line along the Aegean Sea, Dardanelles, Sea of Marmara, Bosporus, Black Sea, Kuma-Manych Depression, Caspian Sea, Ural River, and Ural Mountains.
Politically, the most important consequence from this definition is found in the context of the enlargement of the European Union. Many countries to the East and South of this bloc aim to join, as they see it as providing prosperity, jobs, migration opportunities and political freedoms. The Treaty on European Union states that "Any European State which respects the principles set out in Article 6(1) may apply to become a member of the Union".
The term "European state" was used to reject the application of Morocco as only "European" countries can theoretically join the EU. Interestingly Cyprus, by most definitions, is technically in Asia yet was allowed to join, and Greenland, technically in North America, was for a long time a member as part of Denmark. Turkey was accepted as an applicant as a corner of the country - Eastern Thrace - is technically part of Europe.
The Council of Europe has a similar definition in its constitution and currently includes nearly all "European" (or partly European) countries.
By contrast, Eurovision song contest eligibility is more liberally defined to include the area of the European Broadcast Area including many countries that are not considered part of Europe including Israel, Morocco and Tunisia.