A plane is a flat surface that has length and width, but no thickness. Surface is defined as the exterior of an object. Surface has length and width. You can see it or imagine it, but because it has no thickness, you can't hold on to it.
Take for example the surface of a table top. You can see that it has length and width, you can even run your hand over it, but you can't pick up just the surface. It has no depth. That's the way a plane is. The difference between the table top surface and a plane is that the table top is not perfectly flat and a plane always is. The only time (in this world) we can have anything really perfect is in our imagination, and that's where a plane is. You have to imagine a perfectly flat surface that has length and width, but no thickness.
Of course, to explain or work with ideas using the term plane, we must imagine objects with similar characteristics as planes, even though they may not be perfect. For instance: a table top, a blackboard, or this piece of paper may not be a "perfect" plane. Nevertheless, we can think of them as planes.
Parallel lines are always the same distance apart and always in the same plane.
These lines are parallel to each other because they are on the same plane (the surface of this piece of paper) and they are the same distances apart.
Most materials used in the trades have opposite sides that are parallel. A board of lumber is usually rectangular or square, and each edge is parallel to the edge opposite it. The opposite sides of I beams are parallel. Paper has parallel sides.
If you mark two straight lines down opposite sides of a pipe, you have drawn parallel lines.
Note: This is not to say that you cannot draw lines that are not in the same plane and not parallel, but the lines we draw in our work usually are.