To “launder” money is to make “dirty” money (obtained from illegality) “clean.” For example, illegal drug money might be “laundered” through a legal business, such as a restaurant, creating a perception that the money was legally earned.
 
“Money laundering” has existed in the United States since at least the first half of the 1900s (after the introduction of the income tax). The term “launder,” however, was popularized in 1972 when Mexican bank accounts were used to hide the identities of political donors.
 
   
Wikipedia: Money laundering
In the past, the term money laundering was applied only to financial transactions related to organized crime. Today its definition is often expanded by government and international regulators such as the US Office of the Comptroller of the Currency to mean any financial transaction which generates an asset or a value as the result of an illegal act, which may involve actions such as tax evasion or false accounting. In the UK, it does not even need to involve money, but any economic good. Courts involve money laundering committed by private individuals, drug dealers, businesses, corrupt officials, members of criminal organizations such as the Mafia, and even states.
 
As financial crime has become more complex, and “Financial Intelligence” (FININT) has become more recognized in combating international crime and terrorism, money laundering has become more prominent in political, economic, and legal debate. Money laundering is ipso facto illegal; the acts generating the money almost always are themselves criminal in some way (for if not, the money would not need to be laundered).
 
In US law it is the practice of engaging in financial transactions to conceal the identity, source, or destination of illegally gained money. In UK law the common law definition is wider. The act is defined as taking any action with property of any form which is either wholly or in part the proceeds of a crime that will disguise the fact that that property is the proceeds of a crime or obscure the beneficial ownership of said property.
 
History
Modern development
Many methods were devised to disguise the origins of money generated by the sale of illegal alcohol. After Al Capone’s 1931 conviction for tax evasion, mobster Meyer Lansky transferred funds from Florida “Carpet Joints” to accounts overseas. After the 1934 Swiss Banking Act, which created the principle of bank secrecy, Lansky bought a Swiss bank into which he could transfer his illegal funds through a complex system of shell companies, holding companies, and offshore bank accounts.
 
In the post-World War II era, legislators found themselves in a quandary as they were confronted with a growing list of commercial, fiscal, and environmental offenses that did not actually cause direct harm to any one identifiable victim; there was no stinking corpse. They decided that confiscating the proceeds of crime would adequately deter potential criminals. Anxious to avoid confiscation, organized criminals now needed to give these huge sums of money—not easily consumed or invested in the legal economy without raising eyebrows—a patina of legitimacy: they needed to “launder” it. Money laundering has been dubbed the “Achilles’ heel of organized crime”, for it compels mobsters to seek out and co-opt established businessmen and women with highly technical know-how and access to legal institutions like banks to launder their plunder.
 
The term “money laundering” does not derive, as is often said, from Al Capone having used laundromats to hide ill-gotten gains. It is more likely to mean that dirty money is made clean. At some point in the process there must be a switch between the two; necessarily the art is to keep that switch hidden.
 
Meyer Lansky perfected a predecessor of money laundering, “capital flight,” transferring his funds to Switzerland and other offshore places. The first reference to the term “money laundering” itself actually appears during the Watergate scandal. US President Richard Nixon’s “Committee to Re-elect the President” moved dirty campaign contributions to Mexico, then brought the money back through a company in Miami. It was Britain’s The Guardian newspaper that coined the term, referring to the process as “laundering.” (See Jeffrey Robinson’s three books on money laundering, The Laundrymen, The Merger and The Sink.)
 
(Oxford English Dictionary)
launder, v.
To transfer funds of dubious or illegal origin, usu. to a foreign country, and then later to recover them from what seem to be ‘clean’ (i.e. legitimate) sources. Also transf.
The use arose from the Watergate inquiry in the United States in 1973-4.
1973 Guardian 19 Apr. 14/2 Suitcases stuffed with 200,000 dollars of Republican campaign funds; money being ‘laundered’ in Mexico.
1973 Publishers Weekly 17 Sept. 54/2 A New York lawyer carrying $200,000 in his camera case to be ‘laundered’ in Switzerland.
1973 J. M. WHITE Garden Game 128 Phoenix is a city where the Mafia is well entrenched; its booming real-estate, building and service industries are ready-made havens for ‘laundering’ the extortion and gambling money from Nevada and California.
1974 Globe & Mail (Toronto) 3 Apr. 1 (headline) Kerr concedes U.S. criminals ‘launder’ money in Ontario.
     
Google News Archive
1 September 1972, Milwaukee (WI) Journal, “Outside Investigator Needed” (editorial), pg. 16, col. 2:
They are worried about the way much of the $114,—- was first “laundered” in Mexico—that is, checks were deposited in a Mexican bank account and converted into dollar drafts so donors could remain unnamed.
   
Google News Archive
13 September 1972, Daytona Beach (FL) Morning Journal, “Money As A Campaign Issue” (editorial), pg. 4, col. 1:
MONEY COULD BE a prime issue in the 1972 campaign, the National Observer notes. Succinctly, one of its top political writers said: “Remember: ITT, Watergate, the dairy lobby, tales of contributions being ‘laundered’ in Mexico, a safe full of $350,000 in cash. All of this, and much more, proves the point.”
     
13 October 1972, Albuquerque (NM) Tribune, “Watergate case compared to capers of GOP-bugging prankster Tuck” by William Steif, pg. B4, col. 5:
“I would have hired an armored truck for $60 in San Antonio,” said Tuck, “and sent it to the ranch to pick up the money. And I would have had a laundry truck following it—to launder the money.”
 
Google News Archive
12 April 1973, Lakeland (FL) Ledger, “Reelection Became Boyish Fantasy” by Russell Baker, pg. 1D, col. 4:
Whether the stachel full of cash, the Mexican money-laundering operation, the Arab bazaar in Ambassadorships—whether these seem to be symptoms of a new low in political rot or merely low comedy in slightly bad taste will depend on the observer’s political bias.
 
Google news Archive
12 May 1973, The Bulletin (Bend, OR), “Adviser Connally leaves Texas law firm to avoid possible conflict of interest,” pg. 7, col. 6:
Newsday said that he identified the money as his own “personal” contribution after federal investigators uncovered the Mexican “money-laundering” scheme.
 
OCLC WorldCat record
Dirty money : Swiss banks, the Mafia, money laundering, and white collar crime
Author: Thurston Clarke; John J Tigue
Publisher: New York : Simon and Schuster, [1975]
Edition/Format: Book : English
 
OCLC WorldCat record
Investigation and prosecution of illegal money laundering : a guide to the bank secrecy act
Author: Charles W Blau
Publisher: [Washington] : Gov. Print. Off., 1983.
Series: (Narcotic and Dangerous Drug Section monograph) 
   
Google Books
Practical Investigation Techniques
By Kevin B. Kinnee
CRC Press
1994
Pg. 219 (Chapter 17 Money Laundering):
To understand the process, one must understand the meaning of the term money laundering. The U.S. Justice Department definition is, “The process by which one conceals the existence, illegal source or illegal application of income, and then disguises the source of that income to make it appear legitimate.”