You may think this is a basic question, but there's a lot to correctly hanging a picture from choosing the right hanging hardware to correct wall placement. 
DETERMINE THE TYPE OF FRAME
When it comes to the act of hanging a picture on the wall so that it is secure and level, you have a few options.
First, take note of the hanging attachment on the back of the frame. Is it a sawtooth hanger or is it hooks that need wire strung through? Generally, larger heavier frames will have the wire option.
MARK THE SPOT WHERE YOU WANT TO HANG
Once you've determined where you want your frame to hang, mark the spot lightly with a pencil. I almost always hang my pictures with picture hanging hooks. I only discovered these little guys a few years ago and they have changed my life - seriously! Before, if I used to bang these huge nails into the wall. Half the time the pictures would fall off or if I decided to remove the frame at some point in time, I was left with a large hole in the wall that needed to be patched.
These picture hanging hooks use tiny little nails, are less than 2 inches long but can support up to 50lbs of weight (depending on which type you purchase) due to the fact that the nail goes in at an angle. Genius! Now my pictures hang securely, the nails come out easily and leave only the tiniest hole and I don't have to worry about drilling massive anchors into my wall, either
HOW TO ARRANGE PICTURES ON A WALL
- For most homes, you can hang art at eye level. Unless you're really tall, your ceilings are only 7 ft OR your ceilings are vaulted. Then you can break the rules!
- One of the things I often see is too-small art. If you have tiny piece of artwork that you are dying to display, you may need to display it as part of a gallery wall.
- Orient your art with the furniture over which it is displayed. If you are hanging a frame over a dresser then your frame should be at least three-quarters as wide as the piece of furniture and about 6 - 12 inches over the furniture.
- If you are hanging a gallery wall, lay the artwork out on the floor first, measure the total groupings height and width and make sure it follows the same guidelines above for filling up your space.
Examples:
Judging this gallery wall based solely on height, you can see that a whole extra row needs to be hung closer to the couch. There's too much space between the artwork and the piece of furniture.
Do you see how much better the artwork looks hung lower to the couch? Also notice how the art extends the majority of the width of the couch, but just slightly narrower.
WORKING WITH TALL CEILINGS
Embrace the space and hang larger pieces of art. There's nothing worse than beautiful elegant tall ceilings but tiny artwork on the walls!
Don't be afraid to stack art vertically. You need to visually fill some of the space. With tall ceilings it's better to go up then go out.
What height are the other elements in your room? Pay attention to - and take a cue from - the height of your windows. What you hang on the wall can be as tall as they are:
Take a look at this living room (image provided by a reader) - notice how vast the walls seem and how dinky the fireplace looks in contrast?
When you have vaulted ceilings like this, you definitely don't want to hang a few pieces at eye level. This will leave way too much bare spare above the pieces and make your room feel completely disproportional.
There's a myriad of ways to correctly fill up this space. Personally, I need symmetry in my life so I would opt to balance both sides of the fireplace similarly.
This could be done with shelves that hold frames and other decorative items, with a large mirror or piece of art directly over the fireplace. 
Notice how the shelves go as high up as the adjoining wall on the left. This wall provides a natural visual line across the top of the wall so it makes sense to work with it.
On the other hand, you could group pictures on each side with a larger piece centered over the fireplace:
Finally, go vertical in the center then balance out the sides with single, larger pieces. 
Of course, built-ins would also look amazing to flank a fireplace like this - but those are not in most people's budgets!
HANGING A GALLERY WALL
Gallery walls are a visual balancing act. This infographic should help you get started:
Looking for more tips? Martha Stewart has some great ideas on how to arrange an eclectic group of prints.
Now that you know exactly how to hang a picture, I know you'll get it right each and every time!