Our current representative election system's goal is to make sure that each geographic region's population to have an equal say in politics. But a Proportional Representation (PR) voting system can be even more powerful than this, allowing PR for not only geographic populations, but literally any subpopulation that exists. Eg. if the voters consider abortion to be the most important issue, PR makes certain that the percentage of pro-life representatives matches the percentage of pro-life voters (& the same with pro-choice), as it should be.
PR provides the best protection against gerrymandering.
FairVote has already put forth suggestions as to how a PR voting system would be structured in the U.S. and how the transition will be made.