Definition: Order 66 was the order Chancellor Palpatine gave the Grand Army of the Republic in Episode III: Revenge of the Sith. It was one of several contingency orders given to the Clone Troopers, which they were trained to follow without question in case of an emergency. Order 66, which could only be executed on Palpatine's direct command, called for the Clone Troopers to kill their Jedi leaders. Ostensibly in place to prevent the Jedi from turning against the Republic, Order 66 was really Palpatine's plan to eliminate the Jedi Order so the Sith could take power.
In-Universe: Order 66 states:
In the event of Jedi officers acting against the interests of the Republic, and after receiving specific orders verified as coming directly from the Supreme Commander (Chancellor), GAR commanders will remove those officers by lethal force, and command of the GAR will revert to the Supreme Commander (Chancellor) until a new command structure is established.
(From Republic Commando: True Colors, by Karen Traviss.)
When Order 66 was issued, a number of Clone Troopers believed it to be a false order and began to protect the Jedi instead of killing them. Several other Jedi survived by killing the attacking Clone Troopers.
Darth Vader led the campaign to hunt down and kill most of the survivors in the years following Order 66. This Imperial destruction of the Jedi is known as the Great Jedi Purge. Over 100 Jedi and former Jedi went into hiding and survived the entire Purge; for example, Yoda and Obi-Wan Kenobi survived by going into exile on the remote planets of Dagobah and Tatooine.