Two lines are parallel if they are in the same plane but they can go on forever without ever crossing. You can test this by seeing if you can draw a third line that crosses both lines at right angles (it is perpendicular to them).
Parallel lines can be close together or far apart, but if you connect them with a lot of perpendicular lines (like a ladder), all of the rungs of the ladder will be the same length.
Two planes can also be parallel, if they would never meet no matter how far they extended in all directions. A third plane, crossing both planes at right angles, would be perpendicular to the first two. The opposite sides of a cube are examples of parallel planes.