The video is for the Paul McCartney song "Coming Up," the first single from McCartney's second solo album, McCartney II. From the Wikipedia entry:
"Coming Up" was the opening track from Paul McCartney's McCartney II album, written by McCartney and released in 1980. Like the rest of the album, the song had a minimalist synthesized feel to it. It featured humorously-processed lead vocals from McCartney, who played all the instruments and shared harmonies with his wife Linda McCartney.
John Lennon had liked the song, crediting it for driving him out of retirement to resume recording.
The video is memorable, and it looks like McCartney had a blast making it. Again from the Wikipedia entry:
"Coming Up" is also well known for its video. It is an early example of electronic trickery, with Paul McCartney playing ten roles and Linda McCartney playing two. The "band" (identified as "The Plastic Macs" on the drum kit—a nod to Lennon's "Plastic Ono Band") features Paul and Linda's imitations of various rock musician stereotypes, as well as a few identifiable musicians. In his audio commentary on the 2007 video collection The McCartney Years, McCartney identified the four characters that were impersonations of specific artists:
While others such as author Fred Bronson have suggested that there are other identifiable impersonations in the video, such as Andy MacKay, Frank Zappa and Neil Young, McCartney said the other roles were simply comic relief.
I think it's strange how McCartney doesn't switch over from left-handed to right-handed guitar playing for his Ritchie Blackmore imitation like he does for the Hank Marvin one. Weird. And I'd always assumed the drummer was a parody of Steve Gadd or Animal from the Muppets(!). Most people probably thought the Hank Marvin character was Buddy Holly. The saxophonist clearly is Andy MacKay from Roxy Music. I think this dude looks a bit like Teddy Boy-era John Lennon, complete with the sideburns and ducktail:
The video (nice quality version).