Monetary systems are at the heart of macro-economics. You can trace back all economic forms to the monetary system that makes them run. A monetary system deals with the nature of legal currency, the controlling authority of the issuer and the method by which the currency is given value. Put simply, the value and integrity of the currency is the central variable in economic activity and stability.
Standards
All currencies depend on a certain standard by which the tender acquires value. The metallic standards are fairly simple in that you can redeem all currency by a certain amount of the metal, usually gold. Such currencies are highly stable but somewhat inelastic -- they cannot adjust quickly. The alternative to the metal standard is "fiat" money, where either the state or a cabal of bankers decide how much a currency is worth.
Private Control
Someone creates and gives currency a certain "authority;" really, only two options exist here for that "someone." Either the state, or economic elites, issue and control the currency and its value. Modern economies, for better or worse, usually have a fiat currency controlled by a group of bankers. The Federal Reserve system, a group of private bankers that is independent of any government authority, issues and controls the American dollar at a profit. The argument for this type of system is that bankers know what is beneficial for the economy as opposed to the state -- the fear is that politicians would manipulate the currency for political, not economic, reasons.
State Control
In state systems, the government controls the central bank that issues the currency. In places like China, the currency is under state control and its value is based on state decree relative to the world economy. In 1997, when the Asian economies collapsed due to the speculation of George Soros on the Thai currency, the baht, the Chinese yuan retained its value because the state controlled its worth, not the market, bankers, speculators or any other authority. State control permits the government to stabilize the economy and direct investment to areas that need it. Public goods, rather than private goods, dominate monetary decisions.
Rates
One of the central aspects of a monetary system is the "price" of money at any given time. Some systems, like the German, fear inflation more than anything else. Therefore, rates will shift so as to protect the value of the Euro. Since Germany dominates the European Union, or EU, its banking establishment makes certain the Euro retains its value. On the other hand, the American Federal Reserve wants to keep rates as low as possible to encourage investment. "Loose" versus " tight" money is an ongoing debate. If the system is "loose," then money is cheap. Inflation is avoided because the encouragement of investment will increase production and consumption. "Tight" policies value stability over dynamism in their battle against inflation.