"Budget reconciliation" is a process that allows Congress to quickly pass tax, spending, and debt limit laws. Typically the Senate, which is supposed to be the "cooling saucer", allows unlimited debate. This means that Senators can "filibuster" (talk forever to stall) bills and prevent them from passing. Budget reconciliation limits debate to 20 hours and limits the number of amendments that can be passed (although not the amendments that can be proposed). The Senate can only use budget reconciliation once per year for each subject (tax, spending, and debt limit).
Reconciliation was used in the Raegan era to cut spending, in the 80s and 90s to reduce the deficit and reform welfare, and then in 2001 and 2003 it was used to cut taxes, which increased the deficit. The last time it was used was in 2010 to pass the Affordable Care Act ("Obamacare"). In 2007 the Democrat-majority Congress adopted rules to prohibit the use of reconciliation to increase deficits. The Republican-majority House reversed this in 2011 but prohibited reconciliation from being used to increase mandatory spending. In 2015 the Senate's rule against increasing the deficit through reconciliation was repealed.
Bills to be passed using the budget reconciliation process go through the standard committee process. Each committee is given "reconciliation instructions" for their part of the bill. They send their recommendations to their chamber's Budget Committee, which puts everything together for the chamber as a whole to consider. Once both chambers pass it, the President signs or vetoes the bill. Congress cannot override a veto of reconciliation measures.
The "Byrd Rule" lets Senators strike out portions of a reconciliation bill that are extraneous to the main point (budget changes). This is meant to prevent legislators from sneaking other laws in through the reconciliation process that might not pass through standard procedures.