I've heard that a room temperature super conducting material would be a major discovery. How likely is this within the next century and if discovered what would be possible?
As said in the comments, this is a very broad question, so instead of writing a very long post, I point you to a good article titled "Superconductivity and the environment: a Roadmap": http://iopscience.iop.org/0953-2048/26/11/113001 . The article lists a lot of emerging technologies that make use of superconductors.
The applications of room temperature superconductors would be the same as the applications of normal superconductors, but these applications would just be much easier to realize if cryogenic environment is not needed. Many items listed in the article would become preferred over non-superconducting way of doing things if an easy-to-use material with room temperature superconductivity was found.
Since there is no complete theory as for what causes superconductivity in high temperatures, it is impossible to guess when (if ever) a RTS is found. Finding these materials is basically educated guessing an a lot of trial-and-error. It could be that someone stumbles upon such material tomorrow or it could be that room temperature superconductors don't even exist. There is no way to know.
Broadly, all the applications of regular SCs, without the same cooling requirements.
Some popular ones include power distribution (power lines), power generation/conversion (generators/motors), fusion, "portable" MRI, sensors, electromagnetic anything... etc. The list goes on and on. It would probably be limited only by cost.