You'd have to look at the statutes in effect in Georgia in 1821 to understand the process, but an administrator is generally the person appointed by the court to handle an estate; in many states, an executor handles a testate estate (an estate in which there is a will) and an administrator handles an intestate estate (an estate in which there is a will).
Assuming that there was no will, the statute would generally specify a hierarchy for selection of the administrator. While some states require an administrator to be a local resident, many do not and the hierarchy is typically based on relationships to the deceased person. For example, an adult child is preferred to a sibling.
An agent is a person appointed by a person to act for that person. That may be because the person is not physically present, but could even be just for convenience,
That's a general explanation but you really need to read the laws in effect in Georgia at the time of the proceeding for a completely accurate answer.