In the context of Euclidean straight lines it would mean parallel lines. In the context of a curve and a line (or another curve) it would mean the line and the curve do not meet at any point, but not a lot more can be deduced about them.
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In 2D geometry, that is the definition of parallel lines. Two non-intersecting lines are indeed parallel.
intersecting is when the point where two lines cross
intersecting lines are lines that block each other.
CorrectParallel lines as well as intersecting lines must be coplanar (in Euclidean geometry not quite sure about hyperbolic geometry...).Lines in space which neither are coplanar nor intersecting are called "skew"
A shape with non intersecting lines is 100 percent a parallelogram