This song is about a girl who goes into a fantasy world where she imagines the actor Robert De Niro is her boyfriend. The American actor was a strange choice for the British girl group, but this is a strange song. It's very dark with plenty of paranoia, as the girl feels like the world is closing in on her in a way that De Niro's characters did in movies like Taxi Driver and Raging Bull.
The song was much more somber when it started, as it originally told the story of a girl who gets raped and uses her Robert De Niro fantasy as an escape. They toned it down to make the song more palatable - it was the first single from the album and it needed to get airplay.
Why is De Niro "Talking Italian"? That's because Steve Jolley, who wrote and produced the song with his partner Tony Swain, thought that Italian was a more romantic language, and that De Niro speaking it would make him even more appealing to the girl. The girls in Bananarama collaborated with Jolley and Swain in writing the song, and they hated the line. They let it stand in an effort to minimize arguments - the producers wanted the album to stick with their hit formula, and the girls were pushing to do something different.
Details are sketchy, but according to Bananarama, Robert De Niro asked to meet them after the song came out. The girls had a few drinks before the encounter to build up their courage.
The song was originally going to be titled "Al Pacino's Waiting" before the girls decided that "Robert de Niro" sounded better.