After someone's death, their money and property (called an estate) must be distributed to their heirs. When supervised by the courts, this process is called probate.
Informal probate refers to those estates where the will is uncontested. This method rarely involves a Judge. If there is a dispute amongst the heirs, or the validity of the will, proceeding with the formal probate process may be necessary.
Formal probate is when the requirements for an informal probate are not satisfied, and a Judge is needed to resolve any disputes.