The Great Conversion is a replacement of the old method of database access with a new structured system of database access, object models, etc., into a new package named Doctrine. It involves converting pretty much the entire codebase into brand new code. Along the way, various systems will be upgraded or removed, and some new stuff tossed in. Once Doctrine has been added to a certain system, new features related to that system are much easier to implement, and less likely to be buggy, because Doctrine does all the heavy lifting.
At least, that's my non-coder understanding of how it works.