A discovery was the revelation of something new, previously unknown, or undisclosed. They often came about as a result of deliberate exploration or out of sheer luck. A scientific discovery came about as a result of a scientific study which usually involved the scientific method.
In 2152, T'Pol performed a metallurgical analysis of a 31st century Earth vessel's hull and discovered several unusual alloys, one of which was in a semi-fluidic state. (ENT: "Future Tense")
Based on Neelix's research, "some of the greatest discoveries in Klingon history were accidents. When Sarpek the Fearless unearthed the Knife of Kirom, he was searching for his lost targ." (VOY: "Barge of the Dead") According to Spock, the discovery of the first Slaver stasis box was, "as with a number of discoveries, purely by accident." (TAS: "The Slaver Weapon")
In 2368, according to Captain Jean-Luc Picard, ever since the earliest wormholes were discovered, first year philosophy students were asked the question "What if one of those lives I save down there is a child who grows up to be the next Adolf Hitler or Khan Singh?" (TNG: "A Matter Of Time")
When facing the possibility of becoming stranded in the 20th century, Chakotay considered "pursuing archaeology full time" and "work on one of the important digs in Central America. There were still a lot of important discoveries to be made in this century. I could win a Nobel Prize." ( VOY: "Future's End, Part II")
As a result of the near depletion of resources on his homeworld, Allos told Seven of Nine that "the future of my people depends on [their] discovery" of the Omega molecule. (VOY: "The Omega Directive")