The first Eureka moment was somewhat of an embarrassment. Archimedes, the Greek polymath of the third century BCE was asked by the King of Sicily to determine whether or not his crown was of pure gold, or was an alloy of gold and silver. As with many good ideas, the solution came to Archimedes in the bathtub. He realized that the amount of water displaced by a body was related only to its volume, whereas the mass depended on both the volume and the type of substance of which it was made. Since pure gold is denser than pure silver, the water displaced by 1 kg of gold would be less than that displaced by 1 kg of silver. Since he could find precisely the volume of the crown by placing it in a full bathtub and catching the amount of water that overflowed and could find its mass on a balance scale, he could calculate the precise density and compare it to gold to see if the crown was pure. He was reportedly so excited by this discovery that he ran through the streets (neglecting to put on his clothes first) shouting "Eureka" ("I have found it!").
The term "Eureka moment", then, has come to signify a great discovery in science. While there are occasionally great discoveries, including ones that overturn an existing paradigm, such moments are the exception rather than the rule.