Most commonly, redness and irritation around the fingernails comes from picking at the dry skin of the cuticles. Similarly, biting the fingernails can be a source of the problem, both because the mechanical damage from biting disrupts the cuticles but also because bacteria in the saliva irritated in inflame the skin. Frequent manicures are another common source of cuticle irritation.
I would start by trying to avoid manicures and avoid biting the nails or picking at the cuticles. You can apply a cuticle moisturizing cream to reduce dryness of the cuticles and keep you from being tempted to pick at them.
Occasionally, this redness and irritation can turn into a full blown infection of the tissue around the fingernails. This infection is called a paronychia. If you notice that the tissue is very red and painful, or if there is any discharge from the spot, then you should go to see your primary care doctor
to see if this is what is going on.
Sometimes paronychia will respond to good skin hygiene, soaks in warm water, and topical antibacterial cream. However, sometimes they can be severe enough that they require oral antibiotics or drainage of an area of fluid collection.