This name was not particularly common in Western Europe during the Middle Ages. It became more popular in part due to the fame of Prince Eugene of Savoy (1663-1736), a French-born general who served the Austrian Empire. A notable bearer was the American playwright Eugene O'Neill (1888-1953).
Heinrich was popular among continental royalty, being the name of seven German kings, starting with the 10th-century Henry I the Fowler (the first of the Saxon kings), and four French kings. In France it was usually rendered Henri from the Latin form Henricus.
The Normans introduced the French form to England, and it was subsequently used by eight kings, ending with the infamous Henry VIII in the 16th century. During the later Middle Ages it was fairly popular, and was generally rendered as Harry or Herry in English pronunciation. Notable bearers include arctic naval explorer Henry Hudson (1570-1611), American-British novelist Henry James (1843-1916), American automobile manufacturer Henry Ford (1863-1947), and American actor Henry Fonda (1905-1982).
The name has been very common throughout the English-speaking world since end of the 20th century. In the United States it reached a high point in 2017, when it ranked as the fifth most popular name for boys.