How did Europe's wars of the 18th century affect the American colonies?

Because the colonies were part of the British Empire,
Britain's wars affected them.  The clearest example of this was the French and Indian
War.

This war was part of a broader worldwide struggle
between England and her ally, Prussia, and a coalition of France, Spain, Austria, and
Russia.  This European war had a huge impact on the colonies.  It pulled colonial men
into fighting with the French in North America.  Perhaps more importantly, it led the
government of Britain to try to take more control of the colonies through increased
regulation and taxation.  This was what led to the American
Revolution.

Wars that involved Britain necessarily impacted
the British colonies in North America, with the French and Indian War being the war that
had the most impact.