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 activites they are observing unless asked.
In 2367, Starfleet reached an agreement with the Cardassians that the USS Enterprise-D would take on a contingient 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")