What is the name of the category of viruses that affect only one side of the body?
Short answer: Viruses that affect only one side of the body are not categorized by this information.

Long Answer: For people who likes details

Several types of virus classification exists: 


Baltimore: Classified by the type of DNA or RNA that is carried by the virus
Holmes: Classified by the target organisms of the viruses (plants, bacteria, animals)
LHT: Classified using several physical differences (More specific and complex)


For example, Shingles would be classified as: 

Baltimore: Group I (double DNA stranded viruses)

Holmes: Group III Zoophaginae (animals)

LHT:


Order: Herpesvirales
Family: Herpesviridae
Subfamily: Alphaherpesvirinae
Genus: Varicellovirus
Specie: Human herpesvirus 3 (HHV-3)