A class is a group of people that share a position in society. So, for example, a society might have lords and serfs. The lords (or perhaps more precisely the nobility) are a class. The serfs are a class. For Marx, classes tend to (try to) act in their own interests (at least eventually), and they do so in opposition to other classes; those other classes have opposing interests. This is called class antagonism or class struggle. To be clear, for Marx, this antagonism is a defining characteristic of social classes. A peasant revolt that demands the cancellation of all debts and a redistribution of the land is an example of class struggle. And so is the use of debt and property ownership to control the peasantry. Of course there is the familiar analysis of the proletariat and bourgeoisie; the bourgeoisie struggles to maintain its power over the proletariat while the proletariat struggles to survive, improve its lot, coordinate with each other, and overthrow the control of the bourgeoisie.
A classless society, for Marx and Engels, is a society without class antagonism or class struggle because the possible sociological groupings you might try to use to analyze the society would not be ones that contend with each other; while you could probably still sort people into categories, those categories would not have the quality of having opposing interests that result in a degree of hostility.
A stateless society is one without an authority that relies on monopolizing the legitimate (whatever 'legitimate' is supposed to mean) use of force. A stateless classless society combines these two, though it is hard to imagine having a state without having some degree of class antagonism.