By what steps and contracts does new money enter the economic system? Like many noneconomists who read about economic history and finance, I always remain somewhat baffled by the opaque generation of currency under the federal reserve system. My understanding is that the banks "loan" money to the government, which prints a federal reserve note or IOU signed by Treasury. But then what?

How do the banks actually receive federal currency and/or credit... and determine the amounts they receive? My understanding is that banks can effectively receive as much money as they can "justify" by credit contracts, ratings, and a given ratio of leverage or fractional reserve. Apart from market forces and assumed prudence, are these the only factors that legally limit the amounts of money banks can receive? Can banks essentially get as much money as they say they can securely loan?
The phrasing of your question (the same is true for your description of your understanding) suggests to me that what you need to understand about money creation in banking is much more basic than "steps and contracts". As a full treatment of money creation would not be appropriate on a Q&amp;A site (it'd be really long!), I will instead recommend that you first read about the different types of money (specifically, the difference between base money and broad money), and then that you read "Money Creation in the Modern Economy", which will answer your question about how banks create money. 

The short answer to your question is that banks don't "receive" money to loan out as a part of any formal arrangement, rather, they are allowed to lend out a large fraction of the money that is deposited with them. Importantly, when they give someone a loan, the money that is loaned will tend to be deposited at a bank (because banking is a mostly closed system), which gives the bank that receives the deposit the option to loan out more money. That's how money creation happens— loans that, through the magic of double-entry accounting, result in the bank swapping one kind of asset (reserves) for another (an outstanding loan), while the borrower suddenly has a new liability (the loan they owe to the bank), and a new asset (cash). 

When we talk about money creation, we are not saying that banks literally create money in the sense of asking that some be printed up for them, it's that the banking system acts in a way that generates total liabilities (or claims on others) that are much greater than the monetary base. Because all money is really just claims on others, this is, in effect, money creation.