In short, a Letter of Authorization, better known as an "LOA", is a document from the ATF that allows an FFL to continue to conduct business operations while their renewal application is being processed by the ATF. An LOA is only valid for up to 6 months past their current FFL expiration date.
This form isn't always necessary but it is very important if the renewal process has gone past the FFL holders expiration date. If it has then it is required for you to get and keep a copy of their LOA for your records.
With an LOA you can continue to acquire and dispose to an FFL holder for up to 6 months past their FFL expiration date.
When you acquire from or dispose to an FFL holder with an LOA you should continue to use the expiration date on their current FFL, even if it's expired. You should make a note on the acquisition or disposition when acquiring or disposing to an expired FFL operating on an LOA.
You can see an example of an LOA and read more at the ATF's website here.