Differentials here may include dishydrosis and a fungal infection. Fungal infection involving the finger nails is referred to as onychomycosis and they may cause some changes on the finger nails. If discharge is present in the area, then you may have the discharge cultured. The results of the culture will help determine the causative organism. This will help guide treatment later on. Applying topical antifungal cream with corticosteroid preparation will be able to help. Discuss this with your physician so that the right medication may be given early on.
An infection that develops along the edge of the fingernail or toenail is called a paronychia. It is the most common hand infection.
The most common causes are bacterial or fungal infections which enters the skin around the nail that has been damaged by trauma, such as nail biting, finger sucking, dishwashing, or chemical irritants.
The yellow-green color, indicates that a collection of pus has formed under the skin (called an abscess). If this is the case it may require an incision and drainage.
Warm soaks in a mixture of 50% warm water and 50% liquid antibacterial soap 3-4 times daily for about 15 minutes.
In extreme cases, infection can move under the fingernail and need partial or complete nail removal. In your case there there does seem to be some cellulitis, involved and probably requires antibiotics.
It is important to follow up with your doctor in 24-48 hours to be sure that the infection is healing properly.