The line is defined, by Euclid, as "that which lies evenly with the points on itself." This is utterly useless as a definition, as it seems complete nonsense unless one already knows what a line is.
One common alternative way of describing a line is as the shortest distance between two points, or a geodensic. Now, as explained earlier, the shortest distance between two points on a sphere is along a great circle. While lines on flat planes appear straight, on elliptic and hyperbolic planes they are curves. It follows that while parallel lines on flat planes will remain the same distance from each other for their entirety, on curved planes they will curve towards or away from each other.
In an elliptic plane, such as on our handy dandy sphere model, there are no parallel lines, as they will always curve towards each other. Two great circles of a sphere cannot help but intersect; no matter which line of longitude you choose, it will always have to cross the equator. While the parallels of latitude do indeed remain equidistant from the equator all around, small circles are more equivalent to a parabola on a flat plane than a line. Line segments are given in degrees of the central angle intersecting them.
Similarly, geodensics of a hyperbolic plane will always curve away from each other. On a hyperbolic plane, there are two lines through a given point A asymptotic to a given line L. These are called the "left hand" or "right hand" parallels, depending on which side they converge to line L at (see Fig. 1). The acute angle formed by the parallel and a line perpendicular to line L is called the angle of parallelism, and fluctuates depending on the parallel's distance from line L and how curved the plane is. Lines with a greater angle of parallelism than the asymptotic parallels are called ultraparallels, and will not intersect line L, even unto infinity. Likewise, lines with a smaller angle of parallelism will intersect line L sometime before becoming infinitely long.
^Figure 1