The resignation of Barry McElduff means there will be a by-election in West Tyrone constituency. How does an MP resign and what happens next?
The first thing to note is that MPs can’t resign. A 1624 resolution prohibits MPs from doing so because “a man, after he is duty chosen, cannot relinquish”.
In those days being an MP was unpaid and seen as an obligation, rather than a sought after position.
For an MP to leave the Commons they have to be appointed to “a paid office of the Crown” and therefore become ineligible to sit in the Commons.
Two sinecure positions - offices that no longer have any duties - are maintained for this specific purpose.
They are Crown Steward and Bailiff of the Chiltern Hundreds and Crown Steward and Bailiff of the Manor of Northstead.
The two offices are alternated, allowing two MPs to resign at the same time.
An MP who wants to resign applies to the chancellor of the exchequer for one of the two sinecure positions. They retain that office until someone else wants to resign.
Tristram Hunt is currently Steward of the Chiltern Hundreds and Jamie Reed is Steward of the Manor of Northstead.
The chancellor then signs a warrant appointing the MP and issues a public notice. After the Speaker has been notified, the appointment is noted in Votes and Proceedings.
Side note: As a Sinn Fein MP, McElduff has never taken the oath but is still required to go through this process.
Gerry Adams wrote to the Speaker when he resigned in 2011, which caused some controversy and lots of points of order. The letter was viewed as a request and he was appointed. 
The seat is now vacant. A writ has to be moved for a by-election to take place.
Requesting the Speaker have a writ issued is usually done by the MP’s party. As Sinn Fein don’t take their seats, this will be requested by government chief whip.
The writ, like all election writs, is drawn up by the Clerk of the Crown in Chancery, a senior civil servant who is head of the Crown Office at the ministry of justice. 
The election writ is then sent to the constituency’s returning officer and a by-election is held.
And that’s how MPs ‘resign’ from the Commons.
FYI the chancellor can in theory deny an MPs request but that hasn’t happened since 1842.
This 2010 @HouseofCommons factsheet has many interesting facts about the history of the process parliament.uk/documents/comm…
