Remove the top. Using a pairing knife cut off the top of the pomegranate (the pomegranate's blossom end, the one that looks like a crown/flower).
Score the skin. Holding the fruit you will notice about 5-6 ridges on its surface (it's not perfectly round, the exact number of ridges can vary between pomegranates). Make shallow cuts along the ridges of the pomegranate down toward the bottom of the fruit (from blossom end to stem end). Try not to cut the seeds (pomegranate juice stains), just the white part, and the skin.
Break it apart. Use your hands to open the fruit - grip the pomegranate in your hands with your thumbs planted in the center of the fruit. Gently pull from the center outward, separating each section like segments of an orange.
Separate the segments - break the segments apart with your fingers.
De-seed the segments - using your fingers, scrape the seeds from the white membrane. Some people use a wooden spoon to whack the seeds out of the pomegranate. I have never found this method particularly effective so I just do it with my fingers. I've found that many seeds end up bruised. If you are concerned about staining your clothes or countertop, you can remove the seeds underwater, but you shouldn't need to if none of the seeds have been damaged.
Separate the white pith from the red seeds. Fill a large bowl with water. The seeds should sink to the bottom of the bowl and the white pith/membranes should float to the surface. Discard the white parts using a small sieve or your hands.
Strain the seeds.