While these aren't my favorite 80s songs , I like these two and for me they sum up the whole (early) 80s look and feel... And also when videos told a story -- that may or may not have had anything to do with the lyrics!
I was focused pretty squarely on 60's and 70's music (especially Rush, Yes, Genesis, etc) in the 80's (despite the 80's being my high school and college years). However, there were a number of things I liked, especially the quasi-techno/pop stuff like this:
Erasure - I always thought they had very well written songs, and some of the catchiest pop stuff ever: