Justin's recommendations are good but just touch the surface of a deep literature on the significance of class. Unfortunately, I don't think you'll necessarily be moved either way in deciding whether or how much class matters. The record of empirical research is slim compared to the theoretical claims, and the theoretical claims are a tad one-dimensional compared to the historical record of organizing and militancy. And the history becomes ambiguous and inconclusive, compared to the sometimes clarity cum urgency of the present...and so on...
While most of the literature about class generally asserts that class matters now, the so-called proof lays in several works. The pioneering writing of Hans Speier and Lewis Corey in the 1930's combined with C. Wright Mills _White Collar_ signal a turn in social science toward wrestling with the sometime phantom, sometime all too real middling class. If your interest is more contemporary, Ossowski, Braverman, the Ehrenreichs, Poulantzas, Gorz, Parkin, Giddens, and especially Erik Olin Wright pretty much span the spectrum. Wright's work represented an attempt at theoretical synthesis and empirical study. His most enduring contribution was his demonstration that class has as much, if not more so, explanatory value as do more conventional sociological variables, namely occupation. And this was done on mainstream sociology's preferred terrain - statistical analyses. But Wright's accomplishment (and this was early in his career) worked better for explaning income differences than other things - like what class occupants (people) think, act or feel. Class action (beyond lawsuits) and class consciousness have never fared well in Wright's work.
Most of the writings on class get stuck in a definitional quagmire emanating from the Old Man himself and the judgments calls on the continuities and discontinuites over the past century and a half of capitalism.
More importantly, there's a feeling of futility, I often feel, that shadows a lot of the best class analysis traditions: Whither labor process studies? What happens when radical English historians follow a thread about working class history back to the sheep? Althusser made sociological functionalism Marxist which blew-up the New Left Review, made E.P. Thompson vomit, and took two promising English theoriests and made Hindess and Hurst theoritize themselves out of theory, Marxism, and relevance. Meanwhile, Resnick rethought marxism so much that his turn to epistemology made class analysis, if not class itself, a house with no first floor. Not even the stern spanking from Ellen Meiksins Wood could make even the kinkiest theoretician behave.
While it can't be left for you to choose, since that would suggest an untheoretical genuflection to subjectivity... for contemporary relevance, a quick (American) tutorial might wade through Mike Davis' _Prisoners of the American Dream_, Vanneman and Cannon's _The American Perception of Class_, and Martin Oppenheimer's _White Collar Politics_, and William Domhoff's _Who Rules America?_.
As I think about this reading list, I suspect they all shy away from a conclusion they help make but best said by Adam Przeworski: subordinate classes opt for the relatively more certain limited gains in the short term status quo rather than the far riskier, if more significant gains in long-term radical change. Lenin probably said it first, but he made as a throwaway one-liner and a nasty put down to boot. That doesn't count.
Of course, Przeworski represented the moved toward ratcho, the ultimate in theory fashion among those who dislike the French. Rational choice, of course, was the illness that ultimately devoured Erik Wright. At least, thats the theory.