Probate is a court proceeding, just like other law-suits.
It starts with hiring an attorney -- the personal representative hires an attorney to represent the personal representative. If there is no Will (intestate) then the family agrees who will represent the estate.
Then the attorney and personal representative will discuss topics such as
Assets - what is included in the estate
Beneficiaries - who will receive the inheritance - as stated in the Will (or Intestate beneficiaries, if there is no Will)
The attorney will prepare the documents necessary to petition the court.
The personal representative will sign the documents.
Documents and fees will be submitted to the court.
The court (judge) will review, and approve (or request additional information &/or request a bond ) the petition.
The judge will sign a judicial order which is referred to as “Letter of Administration”
Now, the personal representative is officially appointed by the probate court, and will use the Letter of Administration to show authority to get information from financial institutions, etc.