A document issued by the court to prove the legal authority of the person entrusted to deal with a deceased person's estate (called the personal representative).
If there is a Will appointing a personal representative, that individual is called the executor/executrix. The executor will obtain a grant of representation called the grant of probate.
If there is no Will (or if the Will is invalid), the personal representative is called the administrator and applies to the court to obtain a grant of representation called the letters of administration.
If there is a Will, but it does not appoint an executor/executrix (or if the executor/executrix is unable or unwilling to act), the personal representative is also called the administrator and obtains a grant of representation called the letters of administration with will annexed.