"Boyfriend" is a compound from the early 20th century, with its current meaning. "Girlfriend" is older, but originally referred to a woman's childhood female friend, and gained the complementary meaning to "boyfriend" after the term "boyfriend" had developed.

See here and here.

Edit: someone below has the OED as saying that both originally meant "childhood friend of particular gender" from the 19th century, and the current meaning from the turn of the century.

For a parallel usage in a much older context, I can offer that the Latin term "amicus/amica" which is effectively the same as the English "friend" was used 2,000 years ago with the context we would associate with boyfriend/girlfriend in English. In Catullus's poem 43, the poet describes an unnamed woman (identified in other poetry as one Ameana) as the "amica" of the "decoctoris Formiani" -- the (female) friend of the bankrupt from Formiae; he uses the term friend/female friend/girlfriend to mean unmarried lover.

Some original research from Google's ngram viewer.

This makes me think that someone used it around 1960 that greatly popularized it. I can't tell you who that was, but I imagine it had something to do with the rise in rock and roll music.

But the word had been seeping into American popular vernacular since at least after World War 1. Life magazine, for example, has instances of boyfriend dating back to at least 1933 .

Prior to that, there are mentions of boyfriends and girlfriends in legal proceedings and books of etiquette (1922).

But Google books search is notoriously unhelpful, so I can't make a great study. If someone has access to the OED, they could help us.

I do, however, have a theory on the how and the why of it becoming popular. Since this is a top level post, I will put it in a reply.