Maybe not all strictly horror, but a lot of stuff that was at least fairly scary as a kid born in 1980, starting with the kid stuff and moving on to the more serious horror.
Starting off with animated stuff; The Black Cauldron - the movie Disney wants you to forget. No singing and dancing, but a story that's basically about a Lich trying to raise an army of zombies to rule the earth.
The Secret of NIMH - scared a lot of kids at the time, though I personally never found it THAT scary, it's still pretty dark.
Faeries) - this one might be a little hard to find. It used to be on Youtube but it seems to be gone now. That's probably the only cartoon to have ever actually scared me. I wasn't even 1 when it aired but the music when the Redcaps are walking on the stairs and the fight with the Shadow King are, now that I think about it, probably my earliest memories.
There's also a handful of episodes of The Real Ghostbusters that are creepy and worth watching, but only from the first two seasons; the first two seasons were animated simultaneously, and before they got to working on the third season, Parent-Teacher groups got together and complained to high hell and the show got completely dumbed down into mindless kiddy-fodder with absolutely zero scares. Notable episodes: The Boogieman Cometh, Mr Sandman Dream Me A Dream, When Halloween Was Forever, Ragnarok And Roll, Doctor Doctor, You Can't Take It With You, The Hole in the Wall Gang, The Collect Call of Cathulhu, Knock Knock (my personal favorite episode)
Moving on to some more classic kids movies that have some fairly scary bits;
The Dark Crystal - seriously, the Skeksis alone are nightmare-fuel, but then you find out that they prolong their lives by sucking the souls out of little creatures and turning it into a potion to drink..
The Neverending Story - the Swamp of Sorrow pretty much messed up every single child born in the 80s, and the scene with the sphinx statues was pretty scary, not to mention the giant wolf creature..
The Monster Squad - classic Universal monsters vs. a group of young kids, absolute classic!
More into the horror territory:
Poltergeist - was the first horror movie I ever saw, and the first movie my dad rented when we got our first VCR.. He saw Steven Spielberg on the box and figured it would be kid-safe...
House and House 2 - the first is more horror than the 2nd but they're both pretty good movies, not too much on the gory side.
Critters 2 and Ghoulies 2 are both fairly light-hearted sequels to more serious first movies that I watched a lot when I was a kid.