An individualistic society is generally characterized by individualism, as opposed to a society that is characterized by collectivism. Individualistic societies are centered on the individual “self” rather than recognizing and relating to a collective mentality. Individuals who are part of an individualistic society view one another as not closely connected, and place emphasis on personal achievements and goals over that of the group.
Individual societies are generally less homogenous and place importance on personal goals. They also display rational evaluation of both the advantageous and disadvantageous facets of connections with other people. Individualistic societies possess particular facets of communication such as being on the lower end of power-distance society, and also a low-context style of communication.
An individualistic society is usually defined by contrasting it with a collectivist society. In the former, the individual is seen as more important than the group. In the latter, the group is more important. An individualistic society is one in which people tend to think that individuals are more important than the groups to which they belong.
In individualistic societies, the social rules are oriented towards individual rights. People think that it is right for individuals to do what they want to do, regardless of its impact on the group as a whole. If you want to drive a big, gas-guzzling car, for example, it is seen as good for you to do so even if that is likely to pollute the environment and harm the society as a whole.
In individualistic societies people tend to rely on themselves and they tend to think that people who rely on others are weak. This can clearly be seen in the United States where we have much greater respect for “self-made” people than for those who need help to get ahead. We also tend to believe that it is better to be responsible for yourself (for example, by having your own gun to protect yourself) than to rely on others (such as the police). (Of course, not all people in the US share these beliefs, but our society is, in general, individualistic.)
In short, individualistic societies are societies that value the individual. They want people to be assertive, to do things for themselves, and generally to avoid relying on other people as much as possible. They also believe that it is important to place a greater value on individual rights than on the needs of the society (or other group) as a whole.