1. An exclamation of suddenly being successful at coming up with a solution to a problem or discovering something that was not expected.
2. Etymology: borrowed from Greek heureka, "I have found (it)", first person, singular, perfect active indicative form of heuriskein, "to find".

Historical details about eureka and discoveries.

I have found it!

Said to be the exclamation of Archimedes (Greek-Sicilian mathematician , physicist, and inventor, c. 287-212 B.C.) who, while bathing, discovered how to determine the gold content of a crown made for King Hiero II of Syracuse. His discovery of the principle of buoyancy while bathing, helped him in determining the content of the gold and base metal in Hiero's crown. This experiment led to the discovery of the law of specific gravity.

Eureka is the motto of the state of California; based on the theory that gold miners yelled this out when gold was first discoverd there. California was admitted to the Union on September 9, 1850. There is a city in Northern California named Eureka. Incidentally, Ronald Reagan, Governor of California and President of the United States, attended Eureka College—in Eureka, Illinois.

So, since the 16th century, the term has been used as an exulting (joyous) cry of discovery!