Anomie refers to a conflict within a society with regard to its ideologies and norms and its reality. The anomie theory of delinquency argues that anomic societies cause delinquency because of the stresses on individuals that arise from the condition of anomie.
In our own society, we have an ideology that stresses that all people can and must succeed through their own efforts. We believe in the "Horatio Alger" ideal where people can go from rags to riches so long as they are willing and able to work hard. We believe that we are a meritocracy with no barriers to success.
By contrast, people like Robert Merton argue, the reality of our society is that many people do start out in situations where these beliefs are not true. They are still pressured by society to succeed, but they are not given the opportunities to do so that are supposed to exist. Therefore, such people become frustrated and turn to various sorts of deviant behaviors in order to resolve the conflict. One common example would be resorting to theft or drug dealing in order to get the material signs of success.