I could spend all day discussing 80's songs. I am an 80's guy.
Instead I'll pop 10 songs here. Some of the songs that have been mentioned are great and I'll try not to repeat those ones here.
China Girl - David Bowie (actually, that album Let's Dance is one of the most brilliant albums of that decade)
In a Big Country - Big Country (I actually heard this one before I heard ANY U2 song. they were probably my #1 band until U2's UF was released.)
How Soon is Now? - the Smiths (This song still kicks ass today!)
Kiss the Dirt (Falling from the Mountain) - INXS (from my favorite inxs album Listen Like Thieves)
Don't You Forget About Me - Simple Minds
Sweet Dreams - Eurythmics (I think I realised was being "horny" was all about when I first saw Ann Lennox)
In Between Days - the Cure (80's Cure is best!)
She Sells Sanctuary - the Cult (this band opened my heart to the heavier side of rock)
Add it Up - Violent Femmes (this band was my Chilli Peppers side before I got into that band)
She's a Mystery To Me - Roy Orbison (yes, I know the Edge and Bono wrote this, but this one truely feels like its Roy's all the way. It still one of the most beautiful songs even today. Plus, it is kind of cool picking a Roy Orbision track from the 80's. What a voice!)
Bonus: Best 80's Soundtrack - The Lost Boys