A blue slip or blue-slipping refers to two different legislative procedures in the United States Congress.
In the House, it refers to the rejection slip given to tax and spending bills sent to it by the Senate that did not originate in the House in the first place, per the House's interpretation of the Origination Clause.
In the Senate, it refers to slips on which Senators from the state of residence of a federal judicial nominee give an opinion on the nominee.
Blue slip Wikipedia
The Origination Clause of the United States Constitution (Article I, Section 7, Clause 1) provides that the House of Representatives has exclusive authority to introduce bills raising revenue: "All bills for raising Revenue shall originate in the House of Representatives." Under House precedent, this includes bills "raising revenue and appropriating the same" and the House considers itself to be the only proper venue to originate any bills appropriating revenue. The Senate can circumvent this requirement by substituting the text of any bill previously passed by the House with the text of a revenue bill.
When, in the opinion of the House of Representatives, a Senate-introduced bill that raises revenue or appropriates money is passed by the Senate and sent to the House for its consideration, the House places a blue slip on the legislation that notes the House's constitutional prerogative and immediately returns it to the Senate without taking further action. This blue-slipping procedure, done by an order of the House, is routinely completed to enforce its interpretation that the House is the sole body to introduce revenue or appropriations legislation. The failure of the House to consider the legislation means it cannot become a law. This tactic is historically to be of great use to the House and, as a practical matter, the Senate does not introduce tax or revenue measures to avoid a blue slip.
In the Senate, a blue slip is an opinion written by a Senator from the state where a federal judicial nominee resides. Both senators from a nominee's state are sent a blue slip in which they may submit a favorable or unfavorable opinion of a nominee. They may also choose not to return a blue slip. The Senate Judiciary Committee takes blue slips into consideration when deciding whether or not to recommend that the Senate confirm a nominee.
A report issued by the Congressional Research Service in 2003 defines six periods in the use of the blue slip by the Senate:"From 1917 through 1955: The blue-slip policy allowed home-state Senators to state their objections but committee action to move forward on a nomination. If a Senator objected to his/her home-state nominee, the committee would report the nominee adversely to the Senate, where the contesting Senator would have the option of stating his/her objections to the nominee before the Senate would vote on confirmation.
"From 1956 through 1978: A single home-state Senator could stop all committee action on a judicial nominee by either returning a negative blue slip or failing to return a blue slip to the committee.
"From 1979 to mid-1989: A home-state Senator’s failure to return a blue slip would not necessarily prevent committee action on a nominee.
"From mid-1989 through June 5, 2001: In a public letter (1989) on the committee’s blue-slip policy, the chairman wrote that one negative blue slip would be “a significant factor to be weighed” but would “not preclude consideration” of a nominee “unless the Administration has not consulted with both home state Senators.” The committee would take no action, regardless of presidential consultation, if both home-state Senators returned negative blue slips.
"From June 6, 2001, to 2003: The chairman’s blue-slip policy allowed movement on a judicial nominee only if both home-state Senators returned positive blue slips to the committee. If one home-state Senator returned a negative blue slip, no further action would be taken on the nominee.
"2003: A return of a negative blue slip by one or both home-state Senators does not prevent the committee from moving forward with the nomination — provided that the Administration has engaged in pre-nomination consultation with both of the home-state Senators."