A fact-finding mission was a mission tasked with gathering information on a topic by visiting the location where it was taking place or existed.
In 2369, a delegation of Federation ambassadors visited Deep Space 9 with the aim of going on a fact-finding mission to the Bajoran wormhole. The delegation consisted of Lojal (Vulcan), Lwaxana Troi (Betazoid), Taxco (Arbazan), and Vadosia (Bolian). (DS9: "The Forsaken")
In 2375, Grand Nagus Zek sent Quark on a fact-finding mission to the front lines of the Dominion War. In a conversation with Ezri Dax, he questioned why he had been chosen, as he had other responsibilities. He also wondered what he could possibly tell Zek that the Starfleet intelligence reports didn't already discuss. Ezri suggested maybe Zek wanted a Ferengi perspective on the situation, to which Quark replied that the Nagus could send Nog instead. Ezri assured him that it was because the Nagus trusted him. (DS9: "The Siege of AR-558")