Money can be given an Aristotelian causal account of material, formal, efficient and final causes.
What can be money is something with value, that is a store of value. It must be more than this, because laborers have economic value but cannot be money. Money must be something that has value and can be exchanged, namely property. This is the matter of money.
Formally, money is a measure of value in exchange, that is what it is. I say “in exchange” because if men had no need of things they do not possess, money would not exist, because it facilitates exchange. As a matter of justice, money is the measure that is used to equate incommensurately valuable things such as shoes, houses and food.
Money is brought into existence by dependent rational animals, that is animals that need things and have the capacity to create laws and relations. As Joe Carter notes, money is a matter of shared belief or put differently it is a matter of law and not nature.
Finally, the final cause or purpose of money is to facilitate exchange. It is because we need what we do not possess that we engage in exchange and money makes this possible when the objects of desire are not matching in an exchange. For example, I have chickens and my peer has shoes, but does not want chickens. He wants shingles for his house.
Now when economists talk about the money supply they talk about different sorts of things, that is contracts and property which are essentially different from one another. The money supply is broken up into three types (generally): M1, M2 and M3. M1 is currency and demand deposits.
Let us pause and consider demand deposits. As Zippy noted in a previous post, bank deposits are essentially different from currency. A bank deposit is a sort of investment in the operation of the bank. The bank uses the deposited money to operate by investing in other contracts such as loans, because all of the money it has deposited does not need to be returned all at once. They are playing the game of large numbers, which has its risks but allows for some profits.
Fractional reserve banking then seems to be an act of magic because it is poorly explained as depositing the same money over and over again. What is occurring is that you are investing in a bank with the right of a near instant return of the investment after which the bank uses some of that as means of extending credit and entering into contracts with others to produce a profit.
Now, it is true that these assets can be money and Zippy in a different post notes that stocks and all sort of other things are used as money in exchange. However, I would propose that these sorts of property are money only in a secondary and accidental sense. Money as such is a measure and the regular is used to judge the irregular. One may use a stock as money, but its value is irregular and difficult to judge. The way that it can be used as money is that it is already measured by the currency in order to be exchanged. Stock A is known to be worth $50 per share and Stock B is worth $25 per share. We need not trade the stock for money in order to exchange, because we know 1 share of A is worth 2 shares of B.
This would then bring in the question of currency exchanges and how this is determined. Zippy has some notes about the value of fiat currency being determined as a means of tax settlement, but more thinking is needed on this.
In any case, currency seems to be money in a primary sense in that it measure other valuable things, but other things can be used as money in a secondary way as already measured by currency. I can use the random sticks that fall from my tree as measure of length, but this is only meaningful if I know their length by some consistent standard. I can measure time by the motion of my toddler spinning or my praying the rosary, but only as first being measured by something more regular is it useful.
To return to modern economics, the money supply seems to be currency and those assets that are extremely liquid and easily transferred. M2 and M3 are slightly less liquid than M1 deposits and this seems to be the significant difference.