Peeling skin around the fingernails can mean a range of possible causes, from excessive hand washing without good moisturizing, to infection, to a more complex internal disease situation. Without knowing
your medical history or more about the exact description of what is happening (and without seeing it) it is hard to be precisely sure what is wrong.
If by "below the fingernails" you mean, on the tips of the fingers under the white, detached part of the nail, you may have a condition known as "dyshidrosis" which means the moisture balance in your skin is wrong. For this you need a thick hand cream to use at least 5 times per day, and to minimize water exposure.
If by "below the fingernails" you mean, by the cuticle, there could potentially be a low level chronic infection caused by yeast and moisture.
Either way, if you are uncomfortable, it would be a good idea to see a board-certified dermatologist in person to help identify what is wrong and to help you fix it.