Well, no. It *was *the common language of the people, you’re just not going back far enough! Although even in Roman times, there was a difference between what we now know as classical Latin, and common or ‘vulgar’ Latin. (But it was more a difference of register.)
Even in later centuries, any European ancestors who received any education would have done so in Latin, not their national languages (though of course, many fewer people were educated in those days). This remained the case until around the 17th century, and Latin has gradually lost ground to national languages ever since. Exactly which part slipping away depends on what you are looking at: in the 17th century it was replaced for diplomatic correspondence with French, in the early 20th it ceased to be required for high school graduation, etc.
Even today, it remains a widely studied language, comparatively speaking–but one which has been on the decline for a good 3 or 4 centuries.