Perpendicular and Parallel
Perpendicular
It just means at right angles (90°) to.
The red line is perpendicular to the blue line in both these cases:
(The little box drawn in the corner, means "at right angles", so we didn't really need to also show that it was 90°, but we just wanted to!)
Try for yourself (move points "A", "B" or "N"):
Parallel
Lines are parallel if they are always the same distance apart (called "equidistant"), and will never meet. (They also point in the same direction). Just remember:
Always the same distance apart and never touching.
The red line is parallel to the blue line in both these cases:
Example 1

Example 2

Try it yourself:
Perpendicular to Parallel
Question: What is the difference between perpendicular and
parallel?
Answer: 90 degrees (a right angle)
That's right, if you rotate a perpendicular line by 90° it will become parallel (but not if it touches!), and the other way around.
Perpendicular ...

Rotate One Line 90°

... Parallel !

Parallel Curves
Curves can also be parallel when they are always the same distance apart (called "equidistant"), and never meet. Just like railroad tracks.
The red curve is parallel to the blue curve in both these cases:
Parallel Surfaces
Surfaces can also be parallel, so long as the rule still holds: always the same distance apart and never touching.
Lines and PlanesAdvanced Topic: You can also learn about Parallel and Perpendicular Lines and Planes 
Mind Bender
Something that makes my mind bend: we know that if we have two parallel lines, and we rotate one by 90°, they will be perpendicular to each other, right? Well, does the same apply to curves? Can you have "perpendicular curves", by rotating one of them by 90°? I simply don't know, but it is fun to think about.