Two lines in 2-dimensional Euclidean Space are said to be parallel if they do not intersect. In 3-dimensional Euclidean Space, parallel lines not only fail to intersect, but also maintain a constant separation between points closest to each other on the two lines. (Lines in 3-space which are not parallel but do not intersect are called Skew Lines.)
In a Non-Euclidean Geometry, the concept of parallelism must be modified from its intuitive meaning. This is accomplished by changing the so-called Parallel Postulate. While this has counterintuitive results, the geometries so defined are still completely self-consistent.
See also Antiparallel, Hyperparallel, Line, Non-Euclidean Geometry, Parallel Curve, Parallel Postulate Perpendicular, Skew Lines