A curve is a 1-dimensional space. Even though you draw a picture of it in a 2 (or maybe higher?) dimensional space, you should think of the set of points in the curve as being the only points that we care about.

In a closed curve, you can never fall out of the space by moving around in it. In a curve with endpoints, you can fall out of the space by walking across one of the endpoints, so the endpoints are considered to be boundary components.

A closed curve could be the boundary of something, but it has no boundary itself. The boundary of an open curve is its endpoints, a closed curve has no endpoints.

It's important to distinguish having a boundary vs. being the boundary of something, since in some sense the difference between the two is the whole point of homology.

For a one-dimensional shape, a boundary point is one you can stand on and move in one direction, but not in the other. So for an open line segment 0 < x < 1, there is no boundary, since at every point you can move in both directions. For a half-open line segment 0 < x <= 1 the point x=1 is the boundary, since you can move left but not right.

For a two-dimensional shape (or more formally "manifold"), an interior point is one which locally looks like the plane (you can move in all directions), and a boundary point is one which locally looks like a half-plane. All the boundary points taken together will be a one-dimensional manifold _without boundary_, which is pretty neat. For example, take an annulus (a 2D shape like a CD-rom). The boundary of the annulus is two one-dimensional manifolds (the boundary circles), and those circles have no boundary.

The boundary of a line segment is its end points. Intuitively, it a line doesn't have end points, then it's a loop i.e. a closed curve without boundary

Think of a curve as a railroad track, not the area that it encloses. A closed curve has no ends.

The disk that it encloses, indeed, has a boundary (the curve in question).

That's what open and curve mean in topology. For example, the set of x with 0 <= x <= 1 is open, whereas 0 < x < 1 would be closed. It's infamously confusing terminology, since sets can be both closed and open, or neither...