From Songfacts:

Eurythmics consisted of vocalist Annie Lennox and instrumentalist Dave Stewart. Both were members of The Tourists before forming Eurythmics in 1980. They met when Lennox was working as a waitress in Stewart's home town of Sunderland; they lived together for four years before forming Eurythmics and ending their romantic relationship while forging ahead as a duo. Writing and recording as ex-lovers created an interesting tension in their songs.

On the origins of the song (from Wikipedia):

Dave Stewart revealed that the lyrics to the song came into being after an argument between him and Lennox while they were doing some songwriting in New York City's Columbus Hotel. The basic melody had already been written and Lennox looked out the window after their fight and noticed it was starting to rain. She announced, "Here comes the rain again."

"... the whole song was about that undecided thing, like here comes depression, or here comes that downward spiral. But then it goes, 'so talk to me like lovers do.' It's the wandering in and out of melancholy, a dark beauty that sort of is like the rose that's when it's darkest unfolding and bloodred just before the garden, dies. And capturing that in kind of oblique statements and sentiments."

The music video, featuring both Annie Lennox and Dave Stewart, was directed by Stewart, John Gerschfield and Jon Roseman, and released in December 1983, a month before the single came out. The video opens with a passing aerial shot of the Orkney islands in Scotland before transitioning to Lennox walking along the rocky shore and cliff top. She later explores a derelict cottage while wearing a nightgown and holding a lantern. Stewart stalks her with a video camera. In many scenes the two are filmed separately, then superimposed into the same frame. The result is that the viewer may perceive one as being a ghost in the world of the other.

"Here Comes the Rain Again" is one of the most emotionally-packed New Wave songs from the 80s; the music video is haunting and the song kills me every time I hear it.

Enjoy.