A-ha - "Take on Me" - 1985

Another great pop song from the mid-80s. I've tried a million times to hit that high note and failed, but that doesn't stop me from trying again every time I hear the song. The video itself was no slouch either.

Here's an exerpt from Wikipedia's A-ha page:
(With "Take on Me") A-ha became the first Norwegian band to have a number 1 song in the U.S. The song's popularity earned the band a spot on the American television series Soul Train in 1985, making them one of the few white artists to appear on the black music-oriented show. Sales were aided in the U.S. by heavy rotation on MTV of the new music video, which may have been inspired not only by the truck-chase scene from Terry Gilliam's film Brazil but also by the climactic scene from Ken Russell's film Altered States. The video used a pencil-sketch animation / live-action combination called rotoscoping, in which individual frames of live video are drawn over or colored. It became one of the most instantly recognizable and most enduringly popular music videos in the U.S., where it was nominated for eight awards at the third annual MTV Video Awards in 1986, winning six, including Best New Artist in a Video, Best Concept Video, Best Direction, Best Special Effects, and Viewer's Choice.

With "Take on Me" A-ha achieved the perfect storm of breathtaking pop song combined with a breakthrough video and in the process gained a permanent place in the pop culture memory.