(written from a Production point of view)
According to a scene cut from the final version of "Tears of the Prophets", the Ventani system was a system along the Cardassian border which Garak recommended as the first target in the planned offensive into Dominion space. While it was a long way from Cardassia Prime, its second planet was the birth place of Tret Akleen, father of the Cardassian Empire. As such, it was considered sacred ground and Garak predicted its loss would be devastating to Cardassians everywhere, a great propaganda victory. The suggestion was dismissed since the system didn't hold any strategic value, and its loss would only affect Cardassians, not Jem'Hadar. Eventually the participants in the meeting choose to invade the Chin'toka system, as is seen in the episode.
This system was mentioned in the Stellar Cartography: The Starfleet Reference Library ("Federation Historical Highlights, 2161-2385"). It was located in the Alpha Quadrant. The primary was a G-class star. In the late 24th century, it was an independent system.