People were shortening phrases to their initials for about two thousand years before anyone had a word for it.
Rome was shortening its official name, Senatus Populusque Romanus, to SPQR before the Christian era. By the second and third centuries, Christians read the Greek word ichthys, "fish," as a set of initials for "Jesus Christ, God's Son, Savior." The cross carried INRI. Yet English had no general name for the habit until the middle of the 20th century.
The word itself is built from Greek parts. Acro- comes from akros, meaning "topmost" or "at the end," the same element found in acropolis and acrobat. The -onym ending means "name." Etymonline notes it was abstracted from homonym, an older word for two things sharing a name. An acronym, then, is a name made from the tips of other words.
The dating is a little untidy. German had Akronym by 1921, recorded in a Brockhaus reference handbook. The Oxford English Dictionary long gave 1943 as the first English use, and Etymonline still calls it an American coinage of that year. Later digging found English examples from 1940, in a translation of a novel by the German writer Lion Feuchtwanger.
The word arrived just as the habit took off. Wikipedia notes that military acronyms became fairly common during the First World War. By the Second World War they were everywhere, even in soldiers' slang. Before the 20th century, Etymonline says, the practice was rare outside cabalistic writing and acrostic poems.
Purists still argue over what counts. In the strict sense, an acronym is pronounced as a word, like NATO. Strings spoken letter by letter, like FBI, are often called initialisms instead. Etymonline dates that term to 1957 and admits the line is not agreed on. Many people use acronym for both.
Some acronyms lose their origins entirely. Wikipedia gives scuba as an example of a word whose letters most speakers no longer think of as initials at all.