A dimensional shift was a phenomenon whereby an object or person transported from one location to another via folded-space transport, or shifted to a parallel universe on a different dimensional plane.
Ansata terrorists utilized a folded-space transporter called the "inverter" in order to carry out their activities without being tracked to their base. They did this despite the device having caused cumulative damage which warped their DNA, resulting in fatal alterations to the cellular chemistry of anyone who used it. The crew of the USS Enterprise-D was eventually able to track the Ansata's movements through use of an adaptive subspace echogram. (TNG: "The High Ground")
The Gamma Quadrant planet Meridian transited between this dimension and one which intersected ours via dimensional shift, triggered by cascade reactions on the quantum level in their sun. The reactions triggered quantum fluctuations in Meridian's quantum matrix which transported the planet to another dimension where the inhabitants existed in a non-corporeal form. (DS9: "Meridian")