How we cite our quotes: (Section.Paragraph)
Quote #1
[The fundamental] proposition is: that in every historical epoch, the prevailing mode of economic production and exchange, and the social organization necessarily following from it, form the basis upon which it is built up, and from which alone can be explained, the political and intellectual history of that epoch; that consequently the whole history of mankind (since the dissolution of primitive tribal society, holding land in common ownership) has been a history of class struggles, contests between exploiting and exploited, ruling and oppressed classes; that the history of these class struggles forms a series of evolutions in which, now-a-days, a stage has been reached where the exploited and oppressed class—the proletariat—cannot attain its emancipation from the sway of the exploiting and ruling class—the bourgeoisie—without, at the same time, and once and for all, emancipating society at large from all exploitation, oppression, class distinctions and class struggles. (Preface.6)
Here's a summary of the argument known as historical (and dialectical) materialism. It basically says that everything, including tradition and customs, can be boiled down to economic class struggles, or contests between the oppressors and the oppressed. Then it says that these struggles have evolved to the point where the proletarian workforce can only free itself from the bourgeoisie or rich by forever ending all class distinctions, which allow some people to control others. Marx wanted no more rich, no more poor, no more exploitation and oppression, and he believed new traditions and customs would arise from that new communist society.
Quote #2
The history of all hitherto existing society is the history of class struggles. (Section1.1)
People debate what the ultimate source of traditions and customs is. Some say it's religion, some say it's the need for cooperation, and others have different answers. This quotation quickly expresses Karl Marx's view: it's class struggle.
Quote #3
The bourgeoisie has torn away from the family its sentimental veil, and has reduced the family relation to a mere money relation. (Section 1.16)
Marx is saying that instead of families being tied by the bonds of love and affection, under capitalism what you get are relationships of ownership. The traditional concept of the family in 1848 was basically the same as today's: a small group of blood relations dominated by fathers who are married to wives, with the pair of them producing children. Children are seen as being there to take care of their own parents in their old age—sort of like an insurance policy.