Paronychia is a common condition. Pus under a fingernail is somewhat less common. Regardless, these conditions require medical attention urgently. I would strongly suggest that you see your primary care doctor
. Without intervention, both of these can progress to involve the skin, bone or blood and cause serious damage to the the fingertip or even the whole body.
Paronychia is an infection at the side or base of the fingernail. It can occur acutely, and is often caused by a bacterial infection. A longstanding or chronic form of this is sometimes caused by a fungus. In general, this infection can wall itself off and form an abscess
. These variations are important because it guides treatment. If it is an acute bacterial infection then antibiotics can work. However, if it walls itself off then the antibiotics cannot enter and kill the bacteria. In this case it requires a doctor to open the ball of infection. The other variation is a chronic version caused by a fungus that requires anti-fungal treatment. Pus under the nail can represent a paronychia.
Overall, this requires medical attention. Left untreated this could cause long term finger problems. This could develop into a serious problem. I would strongly encourage you to see your doctor urgently.