You want to immediately clean the area in running water for about 10 mins, to clean any foreign objects and damaged cell debris from the wound. Then, after cleaning, which is the most important step, then you can wrap it in some gauze. This is to protect the wound from being banged into objects while sleeping, dressing up, cooking, etc.
In 24 hours, check to see if the area around the wound is swollen or hot to the touch. If it is, then it is infected and you'll need antibiotics (go see a doctor). If you still refuse to see a doctor despite it looking infected, it will probably heal on its own anyway, but it will take a much longer time. If it looks like a clean cut with no swelling and no red skin around the wound (the wound itself will be bloody and red obviously), then it probably isn't infected and will heal on its own.
source: my husband is a doctor and my fingernail got ripped off.