In most circumstances, the executor named in the will takes this job. If there isn't any will, or the will fails to name an executor, the probate court names someone (called an administrator) to handle the process. Most often, the job goes to the closest capable relative or the person who inherits the bulk of the deceased person's assets.
If no formal probate proceeding is necessary, the court does not appoint an estate administrator. Instead, a close relative or friend serves as an informal estate representative. Normally, families and friends choose this person, and it is not uncommon for several people to share the responsibilities of paying debts, filing a final income tax return and distributing property to the people who are supposed to get it.