Earlier I mentioned “the Fed writes a check against nothing”, which I realized is not entirely correct. The Fed writes a check against the Federal Reserve Notes, which are the printed banknotes you currently carry around in your wallet.
These Federal Reserve Notes are liabilities against the assets held by the Fed.
In principal, you should be able to take a $100.00 bill to any Federal Reserve Bank, and demand payment in some Fed assets (for example, redeem your note for gold or silver). In practice, that is not allowed anymore by law.
Basically, the Federal Reserve Notes are nonredeemable, backed by the “full faith and credit of the U.S. government”, unless the Fed agrees to exchange your note for an asset.
For example, the Fed sells treasury securities (or possibly gold) on the open market, which is the equivalent of the Fed redeeming the Federal Reserve Notes.
Think about this in another way, a private securities dealer “deposits” treasury bills with the Fed, which the Fed in return issues to him Federal Reserve Notes as a demand claim against those assets.
Later on, the dealer can return and “redeem” those same notes with the Fed in exchange for some assets.
In theory, the Fed cannot increase the number of Federal Reserve Notes, unless those notes are backed by an equivalent asset.
In a previous era, those assets would be gold. Every time a member bank deposits gold with the Fed, the Fed can issue more gold backed demand notes to the bank.
But since the Federal Reserve Notes are not backed by gold anymore, any increase to the Federal Reserve Notes must be backed up by some asset, usually U.S. treasury securities, or later on a portfolio of risky assets from a failed financial institution.
To rephrase my previous answer: the Fed writes a check against the Federal Reserve Notes to be issued, which are demand claims against the Federal Reserve assets, payable to a security dealer. who then deposits the check at a member bank.
Then once the check settles, the Fed credits the member bank with an equivalent of recently issued Federal Reserve Notes, that came into existence because of an increase in Fed assets. This increase came from the purchase of the treasury securities from the securities dealer.
Having deposited the check, the security dealer can later on withdraw the Federal Reserve Notes and attempt to redeem those notes and demand payment in assets from the Fed.
The Fed will not do it, unless the Fed agrees to exchange some of those assets for those notes.
If the Fed does agree to exchange the notes for some assets, then in essence, that would be a reverse transaction through Open Market Operations, which the Fed simply sells the treasury security for cash to the securities dealer.
That all said, the real question should be this: why would the public accept nonredeemable banknotes?