Agency MBS Purchase
  
First, let's break down an "MBS." MBS stands for "mortgage-backed securities." These are financial instruments, kind of like bonds, backed by people's home mortgages.
The second thing we have to deal with here is the word "agency." An agency MBS is one issued by one of the so-called government-sponsored entities, or GSEs. These are companies that were created by Congress, though not explicitly backed by the government, meant to foster home ownership. There are three of these GSEs: the Government National Mortgage Association, also called Ginnie Mae; the Federal Home Loan Mortgage Corporation, also called Freddie Mac; and the Federal National Mortgage Association, or Fannie Mae to its homies.
So far so good. Agency MBS are mortgage-backed securities issued by one of these government sponsored entities, like Fannie Mae and Freddie Mac. The "purchase" part of the term "agency MBS purchase" should be easy, right? Just buy a few of these bad boys and we're done.
It's a little more complicated, just because "agency MBS purchase" tends to refer to a specific historical moment.
Back in the mid-2000s, the housing market seemed like it would remain strong for the rest of anyone's natural life. In this environment, banks and other financial institutions bet heavily on MBSs. They owned a lot of them, and everyone assumed they were super-safe. Most people would always be able to make their mortgage payments, right?
Unfortunately, the good times didn't last. By 2007, the housing bubble had begun to burst and by 2008, the economy was in full panic. The MBSs became essentially worthless, or at least got to the point where no one would buy them. Because they had bet so heavily on these things, or were in business with people that did, Wall Street banks either went bankrupt or seemed poised to do so.
The government stepped in with a series of measures over the next few years to try to stabilize the market. It's in this context that the big, historical agency MBS purchase took place. The steps the government took to prop up the economy included a program run by the Federal Reserve to buy a large amount of MBSs in order to unfreeze the market for these assets. Ultimately, the Fed's $1.25 trillion MBS purchase program ran from early 2009 until early 2010.