Every time I've heard President Obama say it, he has said something more specific, to the effect of this: he welcomes the rise of a "peaceful and prosperous China, open to the world." This echoes a frequent direct quote from Chinese officials describing how they hope to be seen by the rest of the world.
For economic reasons, many Americans and American companies stand to gain immensely from China's rise as a consumer society (that is, if China is both prosperous and open to the world) by selling luxury goods, by providing high-tech goods and equipment and expertise which China can't make by itself, by catering to Chinese tourists abroad, etc.
On the "peaceful" side, I'm sure that Obama would differ with his definition of this term from China's leaders, though both benefit politically from the rhetorical ambiguity of the word. The USA, as well as many of China's neighbors, would clearly prefer China to be more cooperative and less adversarial in its dealings with other countries in the region in countless ways (fishing rights disputes, maritime border disputes, trade agreements, cyber warfare, One China policy, etc etc).