"The struggle of class against class is a what struggle? A what struggle?"
Class consciousness is the awareness of the existence of different socio-economic classes and, further, realization of the great impact that class has.
Members of "lower" classes often have a greater class consciousness than do their "upper" class counterparts. However, this is not necessarily the case in societies where class hierarchy is a strict and deep tradition. Members of the English aristocracy, for example, were (are?) thoroughly aware of the priviledges of their position and archly disdainful of their inferiors. Japan also has a famously class-oriented culture, with the majority of the population well aware of their "place".
Class consciousness tends to be lowest among highly homogenous societies and communities. The lack of direct exposure to the "other" retards meaningful awareness of living situations which may contrast with your own. The community in which I did most of my coming of age was, by and large, white and wealthy. As such, my fellow high school students were remarkably incapable of empathizing with or understanding the situation of those with a different class or racial background. The nearby community of Lawrence was an object of fear, as it was the residence of not only a large number of working class people, but primarily black and hispanic working class people.
The United States has the majority of its class-related issues obfuscated by race. People of color in the United States are generally less well-off financially than whites; superficially it can seem that, simply, poor=black or poor=hispanic, etc. A more refined degree of class consciousness would make one aware that all poor people in this country have a common ground that goes deeper than racial divides.
Class consciousness among First World countries also suffers from the export of manufacturing jobs, which were the traditional occupation of the working class. Class divides have become coterminous with international borders thanks to treaties like NAFTA and organizations like the IMF, the World Bank, and the WTO. With the industrial working class safely out of sight, its existence can be forgotten. The poles of economic reality appear to be restored: with the figmentary, monolithic middle class on one end and the destitute homeless poor on the other.
Such illusions allow the socio-economic awareness of the majority of First Worlders to be limited to paternalistic charity, with a shake of the head and a coin into the poor box at Christmas.