Why is the skin around my 7 year old child's index fingernail red and swollen?

Infection: This could represent a paranochyia which can be a bacterial or candida infection. If bacterial it can be drained and it will resolve without oral antibiotics. If yeast , treat with topical antifungal along nail fold. These infections can occur together and need to be treated to prevent the infection from spreading and causing damage to the nail.