How can a currency become worthless?

Most currencies are Fiat Money, not backed by a commodity (like Gold and Silver) but backed by the government that printed it.

If the government behind the Fiat currency becomes insolvent, or acquires extensive debt, their currency may be devalued (relative to other Fiat currencies), their ability to borrow money will be affected (lenders will demand higher interest rates because of the risk), and prices will climb as the currency loses value. At present, Venezuela, Argentina, Zimbabwe, and other countries are suffering due to the inflation they are experiencing. 

Some related websites:
What would you do if your money became worthless? – Prosperous Way Down website
A worthless currency – The Economist (Article on Zimbabwe)
The Greek Crisis Explained – (Infographic) – from Gainesville coins
Many people invest in Gold and Silver as a “Hedge” against devaluation. This can be done with collectable coins and jewelry, or bullion. 
Bitcoin to the Rescue
When fiat currency is failing, the citizenry need a mechanism to trade and move money out of the country without fees and taxes; Bitcoin is an obvious choice.  As a “peer to peer” unit of value,  Bitcoin has minimal fees and a global reach.  It is an ideal mechanism to prop up people when their monetary system is in a shambles, and the value of the local fiat currency drops.
 
In Venezuela, Zimbabwe, China, and other countries, Bitcoin is the method of choice for doing business without government oversight, transfer fees, and excessive taxation.  Even in countries with stable currency, Bitcoin is a welcome addition to the mix of fiat currency, gold, and silver, for acquiring wealth and conducting business transactions.
Here’s hoping your value is strong!