I know that skin is an organ, but what inside the body can be affected by skin cancer?

Depends: What kind of skin cancer. Basal and squamous skin cancers are usually local in occurrence and do not spread beyond confines of skin. Melanomas can and will spread or metastasize to other organs and tissues inside the body.

All can be: Melanoma has the ability to travel to any organ in the body and create new tumors.

Numerous organs: Melanoma can be deadly precisely because it travels to other organs in the body.

Local/distant organs: This depends on the type of skin cancer. Basal cell cancer used to be called a rodent ulcer because it destroys local tissue including underlying muscle and bone which looks as of a rodent were nibbling away at your body.Melanoma can spread via the lymphatics to lymph nodes, lungs, liver, kidneys and brain and nearly any other organ.