In America, democracy is defined as liberal democracy to preclude other forms as legitimate. Liberal democracy means theoretically that anyone can contest for political power regardless of political viewpoint. This means that anyone from a monarchist, to a fascist, to religious fundamentalist, to a liberal, to a communist and anything in-between may run in elections. In practice, various devices are used to ensure there is a window of acceptability, and that fire is primarily directed at the left. I used to think at least fascists and monarchists would also be precluded, but I have been shown to be wrong. In this sense, outside of tight parameters, liberal democracy has inherent contradictions that can easily destroy itself and transition to a different point of (usually temporary) stability, such as has been seen in France when the 1st republic became the French Empire under Napoleon, became a republic again, then switched back to monarchy, and so on and so forth.
I'm not sure what the phrase used for what China is doing is (well, there is "Socialism with Chinese Characteristics"), but as an outsider, my understanding is the idea is that a very broad portion of the population should be involved in the communist party and that democracy should be encouraged within a framework of marxism. Nonetheless, several opposition parties are allowed to run in elections though parties too far to the right are excluded. Still, these opposition parties can be fairly critical and given that China is using a significant capitalist mode of production, some elements of capitalist ideology are allowed. It is a little confusing to be sure.