Vanishing lines are imaginary lines used to create accurate perspective in a painting. They are drawn on the top and bottom horizontal edges of an object, along the object and then extended all the way to the horizon line. For instance on a building, there would be a vanishing line along the top of the roof and the bottom of the wall(s). For a window, the top and bottom of the frame.
If the object is below the horizon line, its vanishing lines angle up to the horizon line. If the object is above, they slope down. All vanishing lines end at the horizon line. And vanishing lines from parallel edges on the same object meet at a point on the horizon line.
Whether or not an object has vanishing lines depends on how it's positioned in relation to the horizon line. Edges of objects parallel to the horizon line don't have vanishing lines. (Why? Because they don't recede into the distance and never intersect the horizon line.) For example, if you're looking straight onto a house (so you're seeing one side only), the front face of the building is positioned parallel to the horizon line (and so are its edges). You can easily check if it's parallel by holding a finger along the bottom of the house and another at the horizon line (eye height).
Don't stress if it all seems complicated and confusing. Reading about perspective is harder than seeing it and doing it. "Horizon line" and "vanishing line" is all the terminology you need to implement one-point perspective and two-point perspective. You already know what one-point perspective is; while you may not know that's what it's called, you'll recognize it when you see it...