Under modern House practice and precedent, you write a letter to the executive of your State indicating your intention to resign, and you submit to the House, also by letter, notification that you have sent such a letter, as well as a copy of the resignation letter. You can see copies of Representative Giffords’ letters to Governor Brewer and Speaker Boehner here. Under the terms of Giffords’ letter, her resignation will be effective at the end of the day.
Given the particular circumstances of Rep. Giffords’ resignation, other floor activity also took place today. Several speeches were given on the House floor this morning — procedurally accomplished by Representative Pelosi seeking unanimous consent to speak out of order, and then yielding to Majority Leader Cantor, Minority Whip Hoyer, and Representative Wasserman-Schultz —culminating in Rep. Wasserman-Schultz reading Giffords’ resignation letter in the well, after which Giffords personally delivered the letter to Speaker Boehner, who was presiding in the chair. About 10 minutes later, the letter was official laid before the House, and read once again by the Clerk.
Why does the resignation procedure matter? A few reasons. First and most importantly, the Constitution provides for the filling of vacancies in the House, which can occur by death, resignation, expulsion, declination, or the House declaring a vacancy. Without a formal resignation, the state of Arizona cannot issues writs of election to fill Representative Giffords’ seat, and without a vacancy the House cannot seat a new Representative. Second, the resignation of a Member triggers clause 5(d) of House Rule XX, which instructs the Speaker to announce that the whole number of the House has been adjusted. This is important for determining any numerical threshold that relies on a fraction of the total Membership of the House, such as the Constitutional quorum to do business. Since August 3, the House has had a whole number of 434 (due to the vacancy of the 1st district of Oregon). Upon the execution of Rep. Giffords’ resignation at the end of the day, the number will be reduced to 433.