There are a number of potential answers to a question like this.
Most generally, you could refer to a region of a plane. A region could really be any sort of shape and could be unbounded, you you might impose various conditions on it (e.g. boundedness, compactness, connectedness, convexity, etc).
But even if you want to stick to analogs of line segments, rays, and lines, there are some options. In a certain sense, you can think of planes as a kind of product of two (non-parallel) lines. In the same sense, a half plane would be the product a line and a ray (again, non-parallel, which I'll stop specifying from here). We can consider regions which are products of each combination of line segments (S), rays (R), and lines (L).
* S×S → A parallelogram.
* S×R → Like a parallelogram, but two opposite sides extend indefinitely in one direction, as rays.
* S×L → A region enclosed by two parallel lines.
* R×R → A quarter plane.
* R×L → A half plane.
* L×L → A plane.