A Ponzi scheme is a special kind of fraud. It is based on a fake investment that one schemer (or group of schemers) gets other people to give money to.

In Ponzi schemes, the schemer basically says, "I found a great way to make money fast. The more you give me, the more I can invest in that cause, and the more I can earn for us all". But a Ponzi scheme doesn't actually "earn money". Instead, all of the scheme's money comes from the investors.

Once a Ponzi scheme gets too many investors, the system will always crash. This is because the investors all expect more money than they invested, and they will get impatient.

These schemes always stop one of three ways:
 The schemer runs away with the money they got. This is what schemers try to do.
 The schemer runs out of money; They will be unable to promise money back right away. This is called liquidity, and makes investors panic and demand their money back, often all at once.
 Authorities (or sometimes whistleblowers on the inside) find out about the scheme and stop it.

The scheme was named after a man named Charles Ponzi. He used the scheme after moving from Italy to the United States in 1903. (Ponzi didn't invent the scheme, though. In 1857, Charles Dickens wrote a book called Little Dorrit about a scheme like this. The scheme is a simple idea, and probably very old.) However, Ponzi's scheme was so big that it became the most popular. His original scheme was based on using countries' money-exchange rates to make money, based on international postage stamps. But money soon stopped being invested in coupons, and went to early investors, and a lot to Ponzi himself. 

Ponzi schemes can be run anywhere, even online, and are being run even now.

More examples 
People advertising Ponzi schemes often use impressive words, that are actually very vague. Examples are:
Hedge Futures Trading (taking a good risk)
High-yield investment programs (gives you back a lot of money)
Offshore Investment (makes money easily in other countries)
Schemers often depend on investors not actually knowing about economics. The Madoff scandal of 2008 showed that even sophisticated people like bankers can fall for them. People are fooled by the schemer seeming to have financial skill or reputation. 

Sometimes, the schemers claim that money can only be made if the investment is kept a secret (away from the authorities or public). For example, Bernard Madoff only allowed the accounting firm run by his brother-in-law to perform audits on his "hedge fund", claiming it had to be kept a secret to earn money. 

Since the investment is very vague, not many investors come very fast. But the scheme often gets speed like this:
 An early investor earns money and gets to keep it. It's actually money from a later investor, but the early investor doesn't know this.
 This investor is very happy, and tells other people about the "investment", wanting to help the "investment" make more money for all of them.
 New investors tell more investors, etc.

Schemers often try to tempt the early investors to "re-invest" the money they've "earned". Schemers sometimes send investors "invoices", showing investors how much they've "earned" and how much they'll keep "earning" if they keep "investing".

Schemers also try to make sure new investors can't take their money back, by making the rules tighter. In this case, the schemer will often have a few investors say they kept their money, making the investment still look productive enough to be good.

What is not a Ponzi scheme 
 A multilevel pyramid scheme is like a Ponzi scheme. Both rely on fake financial promises and investments, but there are differences:
In a multilevel scheme, the second "level" of investors find their own investors to make a third "level", each profiting directly from the next level. Ponzi schemes "center" around the original schemer.
Multilevel schemes only brag about the money the investors from getting new investors. This makes multilevel schemes look good to poor people. Ponzi schemes brag about having special connections with hard-to-find sources. This makes Ponzi schemes look good to rich people
Multilevel schemes "crash" faster. This is because multilevel schemes depend completely on finding new victims. Ponzi schemers can just tempt early investors to reinvest the money they've gotten.

 A bubble: A "bubble" is about re-selling. A bubble is when people buy up all of a product they can, to re-sell it all at a higher price, as many times as they can. The bubble "pops" when buyers stop buying the product, and re-sellers are stuck with product they paid too much for. (The product can be anything.) As long as buyers keep paying more, re-sellers can keep making money. Bubbles don't even need central schemers, because people can do this accidentally. (For example, land prices can "bubble" this way. Prices can increase from wanting to build close to big neighborhoods. Once there's no more new land, re-sellers are stuck with their land.) Bubbles are often said to be based on the "greater fool" theory (depending on people who are "fooled" into paying "greater" prices).  But really, according to the Austrian Business Cycle Theory, bubbles are caused by giving loans to buyers for a certain kind of transaction, and in this case would qualify as a Ponzi scheme. In this case, the loan-givers are the schemers, making money from the losing re-sellers.

 "Robbing Peter to pay Paul": This is when people with debt borrow money to pay their debt, and borrow more money to pay back that debt. This is not a Ponzi scheme, because the debt-havers were not promised high returns or anything. Also, the lenders don't always make money.

 Multi-level marketing: Multi-level marketing (MLM) is when companies sell investors things to re-sell directly to customers. Re-sellers can also make money by referring new re-sellers to the company. This may seem similar to a pyramid scheme, but it's not always the same thing. Honest and legal multi-level markets do exist, and a lot of them simply make money by buying in bulk..

Related pages
 White-collar crime

References

More reading

Other websites 
 Ponzi scheme definition with distinctions, history, and other information

Types of crime
Commercial crime