1814, coined by English polymath Thomas Young (1773-1829) and first used in an article in the "Quarterly Review," from Indo- + European. "Common to India and Europe," specifically in reference to the group of related languages and to the race or races characterized by their use. William Dwight Whitney ("The Life and Growth of Language," 1875) credits its widespread use to Franz Bopp.
The alternative Indo-Germanic (1835) was coined in German in 1823 (indogermanisch), based on the two peoples then thought to be at the extremes of the geographic area covered by the languages, but this was before Celtic was realized also to be an Indo-European language. After this was proved, many German scholars switched to Indo-European as more accurate, but Indo-Germanic continued in use (popularized by the titles of major works) and the predominance of German scholarship in this field made it the popular term in England, too, through the 19c. See also Aryan and Japhetic.
Indo-Aryan (1850) seems to have been used only of the Aryans of India. Indo-European also was used in reference to trade between Europe and India or European colonial enterprises in India (1813).