Repairing damage on the floor
Looks like vinyl tile to me (VCT), judging by the straight lines in the upper photo. You're not going to be able to repair that without replacing the tile, and you won't be able to match tile that old very well. This is especially true if it's sheet vinyl. 

I'd try gently tapping it flat with a hammer, and re-wax the floor to fill it a bit. Any other filler will show worse than the damage itself.