(Before you read this summary make sure you read the summary of chapter 1 of the Communist Manifesto)
Chapter 2 of the Communist Manifesto discusses the relationship between the communists and the proletariat. The initial argument presented in the beginning of the chapter is the communists are the purest representation of the proletariat. The communists, argue Marx and Engels, differ from other parties in that they tap into the material course of history and represent in their ideas not some made up ideology (Marx is highly suspicions of ideology) but rather the manifestation of the working of historical direction and its inevitable, rather than just desires, direction.
An important distinction made by Marx and Engels in chapter 2 of the Communist Manifesto is that communism is an international movement rather than national (hence "Workers of the world - unite!). For Marx (and others) capitalism is closely associated with the nation state and the abolition of the former will also include the abolition of the latter.
Chapter 2, picking up from the chapter 1, analyzes capitalism as dependent on private property (see Marx on the structure of capitalism society). The core goal of communism is therefore set at canceling private property, thus bringing about a classless society. Marx and Engels also discuss hired labor and show it exploitative nature, connecting it as well to capitalism and its eventual demise. They even go as far as calling for the abolition of the family since it is also a mechanism of exploitation and a means to capitalism's ends.
The purpose of communism as presented in the chapter to create a society without any divisions, not social, not national and not even between parents and children. Since these goals are highly ambitious Marx and Engels conclude part 2 of the Communist Manifesto with a list of short term means and actions that can serve to move history in the right direction.
Summary of chapters 3 and 4 of the Communist Manifesto