Dominionese was the language of the Dominion, spoken by the Vorta.
Virtual sensory display headsets displayed information in Dominionese. (DS9: "A Time to Stand")
An example of Dominionese, from Federation-Dominion negotiations held aboard Deep Space 9 in 2374, was recorded and analyzed by a group of Augments:
- Eee-ja maa'na hoo'va, baa'li jen'ku'rada sen'to. - Listen file info
- On the balance, we give up more than you.
Here, according to Jack, Weyoun used a passive voice transitive, which indicated that he was making a request, not a statement as the phrase appeared in English. In this case, the universal translator failed to pick up this particular nuance. (DS9: "Statistical Probabilities")
Presumably, the Founders, while in humanoid form, and the Jem'Hadar also spoke Dominionese.
Based on Dr. Bashir's tone, it's possible "Dominionese" is not the actual name of the language.