Yeah, the human brain is absolutely amazing at pattern recognition. We need to be so that we can survive in the wild. A side effect of this is that we sometimes see patterns where they don't exist which leads us to superstitions eg: my team won the big game when I wore dirty socks so I must always wear dirty socks on the day of the big game - ignoring the times when your team won or lost when this condition was or wasn't met. This is called confirmation bias - you suppress the information that doesn't conform to your hypothesis.
In regards to prayer having an effect on sick people numerous studies have been performed and none have been able to reliably show that prayer has any effect on the course of an illness or recovery from an injury. One study in 2006 (The STEP Project) actually showed that people who were recovering from heart surgery had major post-surgery complications (including death) when they were prayed for without knowing it (52%), when they weren't prayed for (51%) and when they knew that they were prayed for (59%). It's been suggested that the experimental group who knew they were being prayed for might have suffered increased anxiety about their recovery, believing that they "needed" to be prayed for.
As for psychics, most of it is clever guess work and a spooky manner. You may have heard of a technique called Cold Reading. This is where a "psychic" (read: person with a lot of experience in deceiving people and reading subtle body language and micro-expressions) will make educated guesses based on nothing more than your physical presence. You can become quite skilled at this with the right training. The psychic will also lead you to confirm or deny what they're saying, giving their performance an air of legitimacy. "I see a man in your life" (how many people don't know a man?) "He has dark or light hair" (50/50 guess) "I'm getting the impression of the letter A" (Oh my god my brother is named Alex, he is a man with dark hair!) "Has Alex passed away recently?" (No, he lives in the next suburb over) "Oh but he's thinking about you!" (No one can confirm or deny this).
There's another technique a lot of TV psychics use called Warm Reading. This is where as an audience member, you have to provide your name and possibly address when you sign in at the studio/soundstage/whatever. Or maybe in the days prior to attending the show you've been visited by someone selling magazines or preaching door to door. The psychic wears an earpiece and a researcher feeds them information based on overheard conversations and prior observations that you don't know they know.
Another great technique that almost all psychics use is Outright Fraud. They plant someone, sometimes multiple people in the audience and lo and behold, they can miraculously tell everything about them based on messages from the spirit world.
Astrology uses the Forer (or Barnum) Effect to give vague personality traits that could be true of any and all people in the hopes that you'll get a hit on something. A lot of internet personality tests do the same thing (see the colour test thread in Relate). Great studies have been done to show that most people if presented with a properly worded but extremely general statement about themselves will identify very strongly with the picture of their character it presents, despite having nothing in common with each other. General statements like "When with your friends you can be lively and energetic but you can be quiet and nervous when in unfamiliar situations" tend to apply to both introverts and extroverts in equal measure.
And finally! Ghosts. People have already brought up the low-frequency sound factor. That's very important but there's also tolerance for ambiguity. Different people have different levels of tolerance for ambiguous information. Is that a branch of a tree scratching on a window in an old house or is it the ghost of a witch coming to get you? Is that the sound of slamming doors or just the sound of water in old pipes? Couple a low tolerance for ambiguity with a tendency to see patterns where they don't exist and you'll probably get someone who thinks that ghosts exist because a few times they got a cold chill while walking through their grandparents creepy old house. Or my mother who thinks that if she prays for the lights to change it's God doing her a solid when she's late when the lights inevitably turn green.
The world is solid and fixed, all the way through. There's no magic, there's no psychic powers, there's no luck (there is chance and probability), there's no ghosts. There is still wonder and beauty and all the good things that we want to see. There is nothing unexplainable. Only things we haven't explained yet.
EDIT: A graveyard makes for great neighbours! Quiet, trustworthy, unobtrusive and a constant reminder of your own inevitable demise to keep you motivated!
Depending on where you live you might occasionally have to put up with goths having sex in the graveyard though. Or more likely taking photographs of each other lounging by headstones at 7pm.