A listening post was a covert operations center for monitoring communications and territory (as of an enemy) between space-faring civilizations.
As of 2367, the Federation-Romulan border was littered with Federation listening posts, one of several measures making it hard for Romulan ships to enter Federation space without being detected even in spite of their cloaking devices. (TNG: "Face of the Enemy")
Before and during the Dominion War, listening posts on the Gamma Quadrant side of the Bajoran wormhole were employed by both the Federation and the Dominion while in control of the Bajoran exit. Before the Dominion fleet came through the wormhole in 2373, there were at least four. However, all of them went silent in the lead up to the fleet coming through. One of the last messages these listening posts intercepted was a Cardassian message, discovered to be an encoded message from Enabran Tain, though Elim Garak initially claimed it was a five-year old planetary survey report. (DS9: "In Purgatory's Shadow", "By Inferno's Light")
The next year, after the Dominion forces occupying Deep Space 9 managed to deactivate the minefield blocking the Bajoran wormhole, the order for the reinforcement fleet to come through was given via listening posts on the other side of the wormhole. After the fleet failed to appear, those listening posts confirmed that it had indeed entered the wormhole and had thus been lost en-route. (DS9: "Sacrifice of Angels")
Late in the war, shortly after Damar announced that his resistance movement had attacked Rondac III, Federation listening posts were able to confirm their destruction. Not long thereafter, a Federation Alliance listening post picked up a Dominion message indicating that they had located Damar's family, presumably killing them. (DS9: "The Changing Face of Evil", "Tacking Into the Wind ")