There are at least three conceptions of the common good. The first is the view of J.J. Rousseau, who is heavily influenced by the Greeks. On his view, there is a general will, which is the will of the community. This will is analogous to the private will, the will of each individual. Just as we want what is good for us as individuals, so the community wants what is good for the community. Just as what is good for Hanno is not necessarily good for his leg (he may have to amputate is leg, for instance, to save the individual), so what is good for the community may not be good for each individual. just as we always want what is best for us, but we may not know what that entails, so the community always wants what is best for the community, but it may not know what that is, either.
This conception is neither socialist, nor capitalist, nor any other 'ist' until more assumptions/beliefs are added. For example, if you believe that the economy of a country requires Capitalism to develop, then Capitalism is part of the good of the community, and desired by the general will. But if you believe that the economy requires Socialist controls, then that is what you think is dictated by the general will. The conception of the common good is neutral, but the details never are.
Another way to conceive of the common good is simply common interest. Here the community as a whole is not an entity that has a good, or wills what is good. Rather, invididuals within the community recognize that they have the same interest. It may be good for you and good for me to keep land undeveloped. Thus, we have a common interest. This does not imply that you and I form a community, and hence it does not require that the good of the community may diverge from our own particular good. This sort of conception is liable to many difficulties, like the free rider problem, or any other where the good of the community clearly does diverge from each individual that makes up the community. Thus, it may be in our interest to make an army for the common defense, and it may be in our interest to include the draft to fill its ranks. But it will never be in our common interest for me to sacrifice myself towards that end, since it is not part of my individual interest to die.
The third conception is that of the Utilitarians. Here, the good of the community is defined by the greatest good to the greatest number of people. 'Good' can be defined in various ways, but usually, some sort of hedonism is assumed: good = pleasure. The defining characteristic of this view is that it is irrelevant how this is distributed in the community. The only thing that matters is the total aggregation of each individual's good. This then is used to justify certain economic systems which, no matter how unevenly, are said to maximize economic production.
When philosophers and politicians use the term, then, it is not clear which conception they are using, and hence what they are saying.