The Austrian theory of money begins with Carl Menger. In his Principles of Economics he develops a theory, based on economic logic and compared to historical and anthropological evidence, of how money originated.
Menger’s emphasis is on commodity money, which Ludwig von Mises later classified as a form of “money in the narrow sense” and which textbooks today call a form of the monetary base. Menger has little to say about credit.
Later Austrians, building on Menger, developed a “monetary view of credit.” That is, they started from money as the basis of the analysis and discussed credit as deriving from money. This is the textbook view among economists generally, not just among the Austrians. One of the consequences of the monetary view of credit is that it tends to see money as the solid but narrow base on which is constructed a huge superstructure of credit. Diagrams often represent the monetary system as an inverted pyramid that looks likely to topple over at any time, leaving a pile of rubble.
The opposite approach is to take a “credit view of money,” as I think Henry Thornton implicitly did in his Paper Credit of Great Britain and as Joseph Schumpeter discussed in his History of Economic Analysis (see in particular part III, chapter 7, section 4). A credit view looks as money as a kind of residual for settling net credit balances where sufficient trust is lacking to roll over the credit. In the credit view, money is the capstone at the top of the settlement pyramid, which is not inverted and is therefore not in constant danger of toppling over. The credit view also implies less suspicion than the monetary view that banks are “creating money” and that fractional reserve banking is somehow inherently fraudulent.
Since Menger there has of course been a huge amount of further historical and anthropological research into the origins of money. The left-wing anarchist anthropologist David Graeber gives an interesting though tendentious account of some of it in his recent book Debt: The First 5,000 Years, using it to criticize Menger.
My sympathies are with the credit view, because credit is both logically and historically prior to money. Logically prior, because it is easier to imagine a society without money (where outstanding credits are extended, transferred, cleared, or written off) than to imagine a society without credit (where every transaction has to involve barter or cash payment at the moment of consummation). Historically prior, because even hunter-gatherer tribes functioned on a kind of credit, where people performed services for one another today expecting something in return in the future. It was not the exact calculation in terms of a monetary unit we now think of in connection with credit, but it did involve exchange of goods over time and was not simply a gift: those who were able to contribute to the tribe’s well-being but refused to do so tended to get expelled.
(An interesting aside: Menger had high regard for John Law’s ideas on the origins of money: see Appendix I of the Principles, page 318 in my book.)