First, let me be clear that when I say a group "wants to" do something, I'm speaking to some extent metaphorically. Some groups explicitly want to do certain things, other groups have the effect of doing something. I use "want to" in both cases. For example, the bourgeois revolution in England was overtly couched in religious language (the established church vs. freedom of worship), and the religious motivations were substantive. But the overall effect was that those supporting the established church supported the feudal aristocracy; those supporting freedom of worship supported the nascent capitalist class.
Second, I will attempt here to be exclusively descriptive; I will analyze the merits of various positions in further posts.
Third, my goal here is taxonomy, not lexicography. I'm not concerned with what various groups call themselves, or describing the rage of political positions held by those who call themselves "progressives", "socialists", "communists", "anarchists" or whatnot.
We can divide people's and groups' political stance into left and right by their attitude towards class privilege. The left, to varying degrees, wants to increase the privilege of working class and decrease the privilege of the capitalist class. The right, to varying degrees, wants to maintain or increase the privilege of the capitalist class at the expense of the working class.
(This analysis is complicated somewhat because some on both the right and left are for or against the capitalist class not because of any deep economic or moral position but because the capitalist class happens to be the ruling class.)
Progressives (a.k.a. reformists or liberals) want to increase workers' privilege within the current social, political legal and economic system. The preferred strategy of reformists is to use state power, the power of the government, to raise the cost of labor power (the minimum wage, improved working conditions, universal health care etc.) or reduce its use-value (the eight-hour day, mandatory vacations, etc.). Commodity relations will then push the price of labor power to its increased cost; the increased cost is mostly consumed, directly or indirectly, by working people.
Revolutionaries (on the left) want to increase workers' privilege by making fundamental changes to the current political system. Instead of using political power to extract specific concessions from the capitalist class, revolutionaries want to demolish the political privilege of the capitalist class and establish the political privilege of the working class. In other words, revolutionaries want to replace the "dictatorship of the bourgeoisie" with the "dictatorship of the proletariat" - or - put the means of production in the hands of the workers.
Revolutionary groups differ primarily by what sort of changes they want to make.
Revolutionary communists want a direct and substantial role of a more-or-less well-defined government in politics and economics. Communists want to create a new state in Lenin's use of the term, i.e. an instrument of class oppression, the oppression of the capitalists at the hands of the workers. It must be noted that the capitalist class naturally sees the erosion of its privilege as oppression; the communist says, "Indeed. We don't have a problem with that."
Revolutionary socialists want a much narrower and indirect role of a well-defined government. They want to see economic and political power, especially administrative power, be broadly dispersed. Socialists also favor a more-or-less "bottom-up" approach to seizing power from the capitalist class, in contrast with the communists' preference for a "top-down" seizure.
Revolutionary anarchists want a more-or-less immediate implementation of a classless society, and the demolition of authoritative coercion; anarchists are opposed to even the distributed authority of socialism. People are adults on the whole, and once they are freed of governmental tyranny, they can deal with the assholes, parasites and crybabies on the basis of individual action or voluntary cooperation.
Of course all sincere leftist revolutionaries want anarchism as the ultimate goal: the "withering away of the state" and the classless society is an essential doctrine of all leftist revolutionaries (those that do not sincerely subscribe to this doctrine are by definition rightists or crypto-rightists); they really differ in how to progress to that goal, in what steps are immediately required.