How does total money in the economy changes?

Fiat money in modern societies is usually printed (minted is the term used for stamping coin) under the authority of a particular institution, usually called Central Bank, Federal Bank, Federal Reserve etc. In different countries there are different rules that apply concerning its function and independence from central government.

Central Banks also are able of conducting monetary policy with various means and measures. In fact, more than anything, a Central Bank is charged with the responsibility of regulating the amount of money circulating in the economy.

A central bank can augment or decrement the amount of money circulating by using (1) a particular exchange rate or by directly influencing (2) money supply.

Examples:

(1) A Central Bank can change the exchange rate with which it loans commercial banks (the ones that the public most often interacts with), thereby affecting their interest in giving loans and/or the public's interest in saving or taking loans. This effectively changes money supply / the amount of money circulating in the economy.

(2) A Central Bank can buy (or sell) government bonds from (or to) the general public. This is called Quantitative Easing. Since that means that either it injects new money into the economy or that it vacuums existing money out of it, it directly changes the amount of money in circulation.

Please note:

There are many more examples to be given and a lot more clarification. Further reading can be done with my textbooks of preference, Mankiw's Macroeconomics and McCallum's Monetary Economics.