First of all, lets try to define what we mean by "class". The simplest way to begin looking it is to look at the two main classes: the working class and the capitalist class. The working class may be broadly defined as those who must work to survive but have little control over the nature of their work or indeed the circumstances of their lives. In todays Ireland, this includes both blue-collar workers (the "traditional" working class) as well as (due to our changing economy) many in white-collar jobs. It also includes the many women who do the unwaged work of child rearing and housekeeping which ensures a steady supply of new workers for the bosses. Those dependent on the welfare state are also part of this class, since their payments are based on either work they have done or their potential to work.
The distinction between the working class and the capitalist class is best described in terms of their ownership of the things which are needed to do work: land, factories, raw materials etc, which we refer to as "the means of production". In a nutshell, the capitalist class own and control these things. The capitalists pass some of this wealth back to the workers in the form of wages and social infrastructure like education, health and other public services; the rest is profit or "surplus value" for the capitalist class. In other words, work generates wealth but this wealth is not controlled by those who do the work, it is controlled by those who manage and direct the workers. Examples of those in this class would be high-ranking businessmen, large landowners and members of the political elite.
The working class far outnumbers those in the capitalist class; under capitalism a very small number of people have most of the wealth and power. The interests of those at the bottom (housing, good working conditions, healthcare etc) are in conflict with those at the top who wish to maintain their privileged position; this is what is meant by "class struggle". Anarchists contend that it is a fundamental force shaping every society in human history.
The divides are not always so black and white; there are those workers who have a higher standing than those lower down the scale. These are the so-called "middle classes", which can be difficult to define as at each end they shade into the class above or below. Examples would be white-collar workers with some degree of managerial power, self-employed businesspeople or farmers. The creation of a "middle class" enables the capitalists to rule more effectively and invisibly; by creating a sort of "class ladder", individuals are encouraged to compete for a better position. The possibility for collective action to improve the position of all workers is thereby ignored. This is quite understandable, as any working person will tell you they wouldn't mind higher wages or better working conditions. The old tactic of "divide and rule" by the capitalist class also serves to pit one section of the class against each other, divided under gender, ethnic or other lines.
Anarchists argue that it is in the long-term interests of all, regardless of their relative position in the current system, to unite against the capitalists. For this to happen, it is necessary that a class consciousness be developed, so that individuals recognise how they are being exploited and realise their collective power to change society. Unfortunately, this does not always come about spontaneously.
Some libertarians argue that focusing on class is unnecessary and means ignoring other forms of oppression such as sexism and racism. Other leftwing tendencies have argued that these are "side issues" to the issue of class. Anarchist communists see all forms of oppression as having a connection and serving to re-enforce each other. Anarchists are not saying that once we get rid of class, that such things sexism and racism will disappear automatically. Neither are we saying that class is the only oppression but rather that it must be a focus if we are to achieve any lasting change beyond token reforms. For example, admitting more women to positions of power in the current system does little to improve life for the majority of women in the working class.
A truly revolutionary movement can only come from the action of the working class as a class. We are the ones who keep society running, it is well within our ability to remove the political and economic apparatus that forces most of the population to work for the benefit of a privileged elite. Our experience of working co-operatively under capitalism prepares us to take control and victories in our everyday struggles to improve working and living conditions teach us confidence.