The Police released five strong albums over a six year period, and are one of the few rock bands who did not release a bad album, in your author's opinion. While I favor the first three LPs, even the weaker Police material is extremely strong. The band successfully combined elements of punk, new wave, traditional rock, jazz and even reggae into their songs. In addition, they became huge stars, and have sold some 50 million units.
"Don't Stand So Close to Me" was off the superb third Police album, 1980's Zenyatta Mondatta. Recording between tours, the album was completed in roughly four weeks. Subsequently, the band has criticized the album as being rushed, but many critics (including the folks at ERV) believe that it is one of the strongest rock records ever made.
The first (rare) video was recorded at the same time as the video for "De Do Do Do, De Da Da Da." The recording took place in late November 1980 in Grey Rock, Quebec. Although the vid was shown on the BBC's Top of the Pops around the same time, it has since become almost totally forgotten.
The second video of "Don't Stand So Close to Me" is the standard video, and it was recorded later. I remember seeing this one on HBO's Video Jukebox, probably in 1981. It features Sting in his Beat (English Beat to Americans) t-shirt.
As many readers will know, the Police never officially broke up, but they simply went their separate ways (for the most part) after their 1984 Synchronicity tour. A 1986 attempt to re-record their greatest hits was aborted and only remakes of "Don't Stand So Close to Me" (released at the time) and "De Do Do Do, De Da Da Da" (rare but released later) were done. They did re-form for a 2007 - 2008 reunion tour, but have not released any new material since 1983's Synchronicity.