The office of Sheriff (or Shire Reeve) predates that of the Lord Mayor by at least 400 years and was in past times known as the Portreeve. Originally a representative of the King, the Sheriffs now support the Lord Mayor and are resident in the Old Bailey during their year in office.
Each year the Liverymen of the City's Livery Companies meet in Guildhall to elect two Sheriffs at an event called Common Hall. The usual arrangement is that one of the candidates for Sheriff is an Aldermen and the other candidates (there can be multiple) are Liverymen. For those Aldermen intending to progress to the estate and dignity of Lord Mayor, a year served as Sheriff is a mandatory prerequisite.
The Sheriffs perform various duties associated with the Central Criminal Court (The Old Bailey) although day-to-day running of the Old Bailey is in the hands of a salaried officer known as the Secondary and Under Sheriff. In particular the Sheriffs entertain the high court judges to lunch every day that the court is sitting and invite guests from across the City’s institutions and businesses to meet the judges.
One duty the Sheriffs thankfully no longer discharge is that of conducting judicial executions in the City. In the past this duty was delegated to a hangman or executioner - an office unknown to the law and paid for his services by the Sheriff.
Both the Sheriffs (Alderman and non-Alderman alike) are required to have a Arms by the time of their presentation to the Queen’s justices at the Royal Courts of Justice in order to receive their Royal Warrants. Whilst the Sheriffs are elected by the Livery, Her Majesty has until mid-September to ‘stay' their appointment. The photo below is the Shrieval badge and chain of Alderman Dr Andrew Parmley (elected 2014) it incorporates his personal Arms with those of various Livery Companies and other organisations with which he is connected.
Note: Sometimes the Sheriff who is not an Alderman is described as the lay Sheriff or non-Aldermanic Sheriff and you will read these terms in many books, although the title is strictly and simply Sheriff whether or not the occupant of the office also is also an Alderman. Since the 1970s the Sheriffs in the Counties of England and Wales have been retitled ‘High Sheriff’ which implies they outrank the Sheriffs in the City of London - they do not!