According to the Mundell-Fleming model, a small, open economy cannot achieve all three of these policy goals at the same time:

1. A fixed exchange rate

2. An open capital market (no capital controls)

3. An independent monetary policy 

in pursuing any two of these goals, a nation must forgo the third. Every course on international finance should conclude with an exercise to prove the International Finance Trilemma. GregMankiw provides the popular review of the "impossible trinity." BradDeLong provides the following concrete examples

Countries on the gold standard (like the U.S. from 1873-1914) chose to have a fixed exchange rate and open capital markets. They did not have independent monetary policy. (The U.S. did not even have a central bank, although the Treasury performed some of a central bank's functions.)

The U.S. today chooses to have an open capital market and an independent monetary policy. Thus it does not have a fixed exchange rate: you cannot take your dollars to the San Francisco Fed and exchange them for gold or foreign currency at a set price.

Countries in the Euro area, like countries on the Gold Standard, have chosen to have open capital markets and fixed exchange rates and thus they do not have independent monetary policies. The European Central Bank (the ECB) sets monetary policy for all countries in the Euro-zone.

China has (roughly) chosen to have a fixed exchange rate and an independent monetary policy. This means that they must have capital controls, which they indeed do. For example, Article 9 of The People's Bank of China Decree [2006], No. 3 states 1: An individual's foreign exchange sales and domestic individual's foreign exchange purchases shall be imposed an annual limit. Within the annual limit, an individual can conduct a sale or purchase business with a bank by presenting valid identity documents; beyond the annual limit, an individual can conduct a current account business with a commercial bank by presenting valid.

The Burda and Wyplosz textbook also provides a nice illustration. What happens if a nation tries to pursue all three goals at once? To start with they posit a nation with a fixed exchange rate at equilibrium with respect to capital flows as its monetary policy is aligned with the international market. However the nation then adopts an expansionary monetary policy to try to stimulate its domestic economy. This involves an increase of the monetary supply, and a fall of the domestically available interest rate. Because the internationally available interest rate adjusted for forex differences has not changed, market participants are able to make a profit by borrowing in the countries currency and then lending abroad – a form of Carry trade. With no capital control market players will do this en masse. The trade will involve selling the borrowed currency on the forex market in order to acquire foreign currency to lend abroad – this tends to cause the price of the nation's currency to drop due to the sudden extra supply. Because the nation has a fixed exchange rate, it must defend its currency and will sell its reserves to buy its currency back. But unless the monetary policy is changed back, the international markets will invariably continue until the governments foreign exchange reserves are exhausted, causing the currency to devalue, thus breaking one of the three goals and also enriching market players at the expense of the government that tried to break the impossible trinity.