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.

Fiat money is a currency without intrinsic value established as money, often by government regulation. It has an assigned value only because the government uses its power to enforce the value of a fiat currency or because the exchanging parties agree to its value.

It was introduced as an alternative to commodity money and representative money. Commodity money is created from a good, often a precious metal such as gold or silver, which has uses other than as a medium of exchange (such a good is called a commodity). Representative money is similar to fiat money, but it represents a claim on a commodity (which can be redeemed to a greater or lesser extent). From Wikipedia

Paper Money Collapse: The Folly of Elastic Money

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. Venezuela is even considering making their own cryptocurrency, the “Petro”, as a solution to their problems.

Some related websites:
What would you do if your money became worthless? – Prosperous Way Down website
A worthless currency – The Economist (Article on Zimbabwe)
11 Times Currency Was Rendered Almost Worthless (and Why) – Ranker
What Happens When Your Money Is Suddenly Worthless? – Cracked (article on India)

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. Gold and Silver bullion – on eBay.