Having healthy nails is basically a matter of keeping them clean and treating them gently.
Hangnails are simply dry, dead bits of skin. Unfortunately, the trouble they pose is the temptation to bite or pull them off, which can result in removing more than just the hangnail and injuring the surrounding skin. The result can be swelling, redness, pain and possibly even infection. The safest way to get rid of a hangnail is to snip it off carefully with clean, sterilized scissors or a nail clipper.
To help prevent hangnails from developing in the first place, keep hands and nails moisturized. Also, avoid biting or picking at hangnails and the skin around nails in general.