How about: Have you ever been awake at night and got the feeling you were not alone?  Ask some kind of question related to the topic to get people thinking about their own experienes.  You can also give a statistic, such as: Almost 47% of Americans say they have seen a ghost.

If you are doing a speech, the suggestion of showing something paranormal as an introduction is an excellent idea.  One classic example is Steven Speilburg's Poltergeist, a film for which you could easily find clips.  If yours is a written assignment, you may wish to research the activity that occurs at Edgar Allan Poe's home.  For, there has been a wide range of paranormal activity associated with the Edgar Allan Poe House Museum in Baltimore, Maryland. Many individuals have contended that an elderly lady haunts this house as an "energy" that lingers in the structure.  For instance, there have been reports of sudden winds that slam doors, and a window suddenly crashing to the floor.  Since Poe is a very popular author, reports on this paranormal activity may interest the audience/reader of your assignment.

I'd start with a definition of paranormal.  That which is inexplicable using science and standard knowledge.  Surely that covers quite a bit, so you can proceed from your definition (use a dictionary if you want as a source) to the examples of paranormal activity you wish to talk about in your essay.

A reader will often be naturally skeptical with this topic, so it's good to use examples they may have heard of before at first, and get to the more unbelievable examples as your paper progresses.  You might also include in your introduction details about those who study and monitor paranormal activity for a living, and how they do that.