I assume that you are using this phrase in the way that Fareed Zakaria uses it...
To Zakaria, the idea of the "rise of the rest" is the idea that the United States should not feel that it is losing its power. He says that Americans tend to feel that they are falling behind and that that is a bad thing.
Instead, Zakaria says, we should feel that we are still doing well but that the rest of the world is rising -- it is doing better economically. He says that we should welcome this. We should understand that this is good for us and good for the world.
As far as what the "rise" is, Zakaria says that it is the rest of the world getting more power in ways other than military ways. He says
along every other dimension—industrial, financial, social, cultural—the distribution of power is shifting, moving away from American dominance.
The previous post was very accurate. The Newsweek link included is also very good. If I may add one more nugget, it would be that the "rise of the rest" is a condition that forces all of us, especially America, to fully embrace the reality of the global world. The perception that used to drive America in that it is the unquestioned leader of the free (and possibly entire) world, is one that is been undermined by the rise of the rest of the world. The globalized and interconnected reality means that America is going to have to share the world with other populous nations that are staking their claim in the new setting. Nations like China, India, and Brazil will have a significant impact on how the future is decided. Whereas America used to set the pace, now other nations will be doing the same. The rise of the rest simply means that the stage will have other players and actors on it, more directors running the show, and more scriptwriters choreographing this little drama. Zakaria's assertion is more positive than anything else. Due to the fact that we live in some of the most calm and bloodless times, it is one where nations have been able to surrender to economic prosperity and growth in order to find their own place in this setting, one where the rise of the rest has helped to define a new and "flat" world.