This is about a man who is pursuing a woman, but the woman doesn't return the interest because she is still in mourning for her previous lover, and hence she always dresses in black. The depressing subject matter is hidden by the upbeat music. (thanks, Andy - Arlington, VA)
There is speculation that the song was written about mourning the loss of Stuart Sutcliffe after he died of a sudden brain hemorrhage. The song was a 50/50 effort by both Lennon and McCartney, but started by Lennon as a response to his own mourning process (which he never really got over). The "baby in black" would be photographer Astrid Kircherr, who dated Sutcliffe before he died.
This song uses a 6/8 time signature, similar to a waltz.
This is one of several Beatles songs with a dual melody line - " If I Fell" is another. McCartney and Lennon sang into the same microphone, making it hard to distinguish which is the main melody line. Sheet music of the song usually displays both. (thanks, Bertrand - Paris, France, for above 3)
This was the first 50/50 Lennon/McCartney song written since " I Want To Hold Your Hand" a year earlier. They wrote it together sitting practically nose to nose at John's Kenwood Estate.