An observer is someone who is tasked with watching and listening to a particular event. Typically this is done for the purpose of ensuring that it unfolds in a particular manner. Observers generally are not expected to involve themselves in the activities they are observing unless asked.
In 2365, Sirna Kolrami boarded the USS Enterprise-D as an official Starfleet observer for the Starfleet battle simulation titled Operation Lovely Angel between the Enterprise-D and the USS Hathaway. On this basis, he attempted to assume command of the Enterprise-D when the two starships were imperiled by the Ferengi, but Captain Jean-Luc Picard nullified his order. (TNG: "Peak Performance")
In 2367, Starfleet reached an agreement with the Cardassians that the Enterprise-D would take on a contingent of observers during its mission inside Cardassian space to locate the USS Phoenix. This was done as a show of good faith. The observers included Gul Macet and Glinns Daro and Telle. (TNG: "The Wounded")
While incacerated in the Federation Penal Settlement in New Zealand, Tom Paris agreed to be brought on board the USS Voyager as a Starfleet observer to locate the Maquis ship Val Jean that had disappeared in the Badlands. He was assured that when it was over, he would be cut loose with a favorable report at his next review. However as the Voyager was stranded in the Delta Quadrant, Paris joined the crew permanently. (VOY: "Caretaker")