The main focus of Durkheim's extensive work in sociology is on the glue that binds societies together. It is this glue—common institutions, cultural norms, shared experiences and perspectives—that makes a society work and which allows people to come together to work for the common good.
To this end, Durkheim developed the hugely influential theory of group solidarity to account for the functioning of modern societies. Such solidarity was achieved through what Durkheim called a collective consciousness, a shared form of thinking that emphasizes commonality in any given society. Group solidarity also came about due to collective engagement in rituals, both religious and secular, that served to remind us of shared interests and values.
Where there is a marked absence of group solidarity, there arises the problem of anomie, a concept developed by Durkheim to show the alienation that individuals often feel from society in the midst of rapid social change. Contemporary society has undergone even more profound and disruptive change than in Durkheim's day, which makes his theories of group solidarity and anomie even more relevant than ever before.
As for Weber, his main contribution to sociology lay in the link he established between capitalism and the Protestant work ethic. From this theory, he also developed the influential concept of the so-called iron cage, which demonstrates the extent to which the individual's life and worldview are irrevocably shaped by the society in which he or she lives.
Society, this great iron cage, imprisons us, forcing us to live out its dictates and reproducing its values, whether we intend to or not. The technological and economic relationships that grew out of capitalist production in turn become fundamental forces in society, conditioning and determining us to act in certain ways.
This theory has become especially influential for anti-capitalist thinkers seeking to provide a radical politics that breaks free once and for all from the iron cage of late capitalism.