Word of caution - pomegranate juice stains. While this method of cutting and de-seeding a pomegranate shouldn't produce much juice, you may want to take care of what you are wearing while cutting or eating a pomegranate.
Make a thin slice on the bottom of the pomegranate (the part without the crown) with a sharp knife, slice off of the stem end of the pomegranate.
You'll then notice gentle ridges along the outside of the pomegranate. Use your knife to cut along those ridges, just through the red part of the pomegranate skin, from blossom end to stem end. If you can't feel the ridges, don't worry about it, just make several gentle cuts from top to bottom.
Once done, use your fingers to pry open the pomegranate. It should open easily, exposing the seeds (more accurately called arils).
It helps to work near or over a large bowl so that as you open the pomegranate, any loose seeds find their way to the bowl.
Pry the seeds away from the peel and membranes.
If you like, you can fill the bowl partway with water. The seeds will sink to the bottom of the water while pieces of the membrane will float to the top, making it easier to separate the membranes from the seeds.
Put the arils into a serving bowl and munch away!
For further reference, check out this helpful video by Food & Wine: