Money as a Medium of ExchangeIn a barter economy, one can only trade goods for other goods. If you want/require something like for instances clothes, and you only have wheat yourself which you cultivate, you need to find someone who can supply you with clothes and at the same time demands wheat.
For a trade to take place, a double coincidence of wants needs to be in place. Trading goods for goods can be limiting as one might have to first trade wheat with someone who has carrots, trade the carrots with someone else for spinach, and then go back to the person with the clothes because spinach is something he/she's willing to trade for clothes.
This type of inefficiency led people in even early primitive communities to come up with some form of money (for instance obsidian) to facilitate the exchange of goods. The advantage of a monetary economy is that the requirement of the double coincidence of wants falls away. As long as the wheat farmer can find someone who wants to buy his wheat, he can buy clothes with the money received for the wheat.
Money then serves as an intermediary to facilitate the process of exchange, making it more efficient.
Money from this perspective is anything that is generally accepted as payment for goods or services or that is accepted in settlement of debt.
What makes money "special" is that it is accepted as payment because people believe that it will be accepted as payment by other people. In England in the 12th century for instance, they came up with 'tally sticks' as a medium of exchange, which was basically a piece /stick of wood with notches in it. The use and exchange of money is thus completely dependent on its acceptance and belief of people as it being 'money' and it being 'valid'/'valuable' -- and so is completely based upon agreement.
Money does not have to be 'backed up' by anything such as gold or silver -- as again gold and silver are simply materials which we've decided to give value -- but that doesn't make them valuable in fact. Backing up money with 'gold' or 'silver' is then simply 'backing up money with another form of money' and does not make the money more 'real' / 'valuable' / 'better'.
Money as a Unit of AccountA unit of account is an agreed measure for expressing the prices of goods and services. In a money economy the prices of goods and services are expressed in monetary terms and so money also functions as a unit of account. The accounting function of money is secondary to that of the function as a medium of exchange.
Also note that money can lose its usefulness as a unit of account when going through a period of inflation, as when prices increase but your income/savings stay the same = you get less for the same amount of money.
Money as a Store of ValueMoney also functions as a store of value. The most common form of holding wealth is money, as it is something which you can easily exchange for something else at any point in time.
There are other forms of storing value, such as property, stocks, shares etc. Of all the different forms of stores of value, money is the most 'liquid' one (since its easily exchangeable).
Holding / storing wealth in the form of money over long periods overtime also has very definitive disadvantages. Overtime, inflation progresses and your money will slowly but surely lose some of its value overtime (and will diminish a lot in periods of hyperinflation). In short, during inflation your money does not retain its value.
n the long run, one is better off storing one's wealth in the form of property, Art, precious metals, etc. As prices increases, the prices of your Art pieces will also increase, and so will retain its value better than money in its simple form.
Money's function as a store of value also implies that it can be used as a standard of deferred payment. This practically means that money also works as the measure of value for future payments.