For the purposes of the discussion on the site, the concept of "civilization" should be reduced to the exchange of information between conscious, self-aware or sentient individuals, which would seem to necessarily result in a accumulation of useful know-how. It is not the intention to think only of a certain kind of existing or "ideal" civilization. One could also refer to "intelligence" rather than "civilization", if that makes things more clear.
When thinking about the survival of civilization we think about civilization as a way of organization that enables intelligent beings to share and use information, collective learning as David Christian called it in his TED talk in 2011. This seems fundamental to gain a degree of freedom from the dictates of natural processes. In maintaining civilization it seems necessary that useful knowledge is maintained through time.
Nature demonstrates effortlessly how the biosphere can survive, evolving over billions of years. To maintain civilizations capable of providing benefit to its members in the long term is the big challenge. History provides examples of civilizations that perished, causing information to get lost, and with it the ability to sustain as many people as lived before in a self-reinforcing crash. Maintaining the ability of collective learning is vital.
Civilization is not limited to the civilization those able to access this site live in. milliongenerations.org is founded on the hope that humans will find ways to continue and evolve civilizations. That may include parallel civilizations existing on the planet at the same time. Both the civilizations and the intelligent beings in the future could be entirely different from the present homo sapiens and today's civilizations. The fewer assumptions are made for the arguments, the more relevant the conclusions will be.
It may be feasible and eventually desireable to spread to other planets and solar systems. This, however, is not the focus of milliongenerations.org, see the motivation page.
Wikipedia has a helpful article on civilization, see here.