Emile Durkheim is often termed as a functionalist sociologist due to his theories on social function. In fact, he is generally regarded as the founder of the functionalist movement in sociology; the modern functionalist perspective is based on many of his central theories. Durkheim was most interested in the "social function" of societal elements—that is, their designated purpose in the society in which they are created. Durkheim saw society as a group of interrelated parts. The example I typically give my students to explain Durkheim's ideas on social function is that of a wristwatch. Any watch is made up of a variety of parts and pieces, each with a particular function. The battery, the hands, the band, and the gears all have their own purpose to serve, and so long as they function correctly, then the entire watch functions. This is just the same when we consider society itself. If all the parts of society remain functional (as opposed to dysfunctional), then society moves along and gets along.
Weber, on the other hand, is often described as a conflict sociologist, due to his agreement with the ideas of Karl Marx and Frederick Engels. Marx and Engels wrote that economic differences are the root cause of many social conflicts (hence their theories in Das Kapital). Weber, who came a bit later than Marx, ultimately agreed with him. However, he went on to say that in addition to economic considerations, any inequality in social structure and political structure would also contribute to social conflict. The basic ideas of Marx and Weber could brand them as the progenitors of modern conflict theory in sociology.