In law enforcement terms, a person of interest is someone that has not been charged with a particular crime, but who is being watched/observed due to the likelihood that the person is complicit in a crime.
The crime can be any type, and once law enforcement establishes a person's guilt, they will charge the person with the crime so that a court of law can decide whether or not the person is in fact guilty of the crime.
If someone is considered a person of interest, it is common that law enforcement will take additional steps to determine the involvement of the person in the particular crime.This might include wiretapping or gaining a warrant to search the person's property (to seek evidence linking the person to the crime).
There is no time limit for which someone is a person of interest, but law enforcement will usually attempt to reach a conclusion at the earliest possible opportunity.