Horizon line is a confusing perspective term because when you hear it, you tend to immediately think of "the horizon" we see in nature. That is, the horizon as in the line where the land or sea meets the sky in the distance. In a painting, the horizon line might be this if you're painting a landscape, but it's best to disconnect the two. Rather, when you hear "horizon line", you want to be thinking "eye level line".
If you draw an imaginary line across the scene at the level of your eyes, that's the horizon line. As you change position, for instance walk up a hill, the horizon line moves up with you. When you glance down or up, the horizon line doesn't move because the level of your head hasn't moved.
The horizon line is an imaginary line used to create accurate perspective in a painting. Anything above the horizon line slopes down towards it, and anything below the horizon line slopes up towards it. Depending on what it is and how it is positioned, this may be very obvious or it may be very slight. Something that straddles the horizon line will slope both up and down. The horizon line is important because the painting's perspective is constructed from this.
Perspective Assignment #2: Spend some time observing how objects are positioned in relation to your eye level, whether they're sloping up or down (or parallel to it). Sit somewhere that's got lots of strong lines, such as a large room with lots of furniture and shelves. Use one finger as the horizon line, and a finger on the other hand to judge the angles of various objects in relation to the horizon line.