Red,puffy, irritated. A paronychia is an infection around the cuticle of the finger or toes. It is usually a strep or staph but could be fungal, viral or any number of other microbes. As it heals, it should lose some of the redness, perhaps be a bit shiny, may be swollen, cracked, tender and may still have some pus draining. Wash well, take all of your antibiotics, use some topical antibiotic ointment and follow up.
Acute paronychia. Acute paronychia is a bacterial infection of skin around nails, usually with staph sometimes other bacteria. Full blown, the area is swollen, red, hot, painful and may feel like there is fluid inside. May be topped by a pustule next to nail. When resolving, the skin looks a little darker, dry, peely and shrunken/wrinkled, mostly because fluid and inflammation are now gone. That will resolve.