The multidimensional transporter device was a device native to this universe that was capable of transporting a person from one universe into another.
In 2371, Miles 'Smiley' O'Brien used the device visit the mirror universe for an unknown purpose. It is possible he was seeking assistance for the Terran Rebellion from Captain Sisko of that universe. (DS9: "Through the Looking Glass")
In 2372, Jennifer Sisko used the device to visit the mirror Sisko and his son, Jake. Jennifer subsequently brought Jake back to this universe, leaving a multidimensional transporter behind as an invitation for Captain Sisko to follow them. Although Sisko attempted to take Major Kira and Chief O'Brien with him, he found that the device was only programmed to allow him to make the transition. While here, he assisted the Terran rebels, who were fighting against the Alliance, to complete construction on the ISS Defiant. (DS9: "Shattered Mirror")
In 2374, Bareil Antos, along with Intendant Kira Nerys, used the device to transport to the mirror universe to steal an orb. Major Kira was eventually able to convince Bareil to betray her counterpart, and abandon the plot to steal the orb.(DS9: "Resurrection")
In the mirror universe, Miles O'Brien somehow obtained schematics for this device, which he subsequently gave to Rom to study. In 2375, Rom left a PADD out containing the schematics; Grand Nagus Zek found the PADD and duplicated the interdimensional transporting process. Zek then used the device, entering this universe in hopes of establishing commercial relations, but was subsequently captured by the Alliance. Regent Worf held Zek hostage, promising to release him to Quark and Rom in exchange for a cloaking device, which did not exist in this universe. Using another transporter device, Ezri Tigan brought the two Ferengi, along with the cloaking device, back to this universe, where they were able to free Zek. (DS9: "The Emperor's New Cloak")