After Marx, many theoreticians from the Left were thinking that once they nationalised the economy, they could subdue the markets and get rid of the economical laws of capitalism. This was to be achieved because, according to them, there weren't many different, private capitals competing with each other...

Let's recall Marx for a while in order to get this sorted out..
Marx says that if there was to be capitalism, there have to be many private, competing capitals i.e. individual capitalists acting in the same markets. Only through this process, through competition capitalists have to act like, err, capitalists. Competition is the reason that forces capitalists to accumulate and improve productivity and so forth. Capitalists cannot use profits for their personal consumption; otherwise they soon are ex-capitalists because others - ones who invest in productivity - will impel them off the markets. It is the relationship between the exploiters that makes capitalism to be what it is.

Now, the economy of Soviet Union should have been non-capitalist economy according to what we have learned from Marx. Still, some leftwingers argue this was not the case. Trotskyists say that we have to put the Soviet Union in the right context and think about its role in the whole world.

The Soviet Union was not the whole world. They see SU as one big factory. Inside the "SU factory" there weren't competing, separate capitals but "SU factory" was part of the world where countries competed, merely in militarian fashion. Trotskyists say that the outcome of military competition was the very same as that of private capitalism. Competition in military sector forced SU to accumulate and invest heavily on arms production.

Whereas Trotskyists favor permanent revolution and world revolution, anarchists and others like-minded don't judge the Soviet Union in terms of military competition but rather in terms of workers' autonomy. For them capitalism and, hmm, non-capitalism are defined by the nature of work. In capitalism workers are told what they should do but in non-capitalist working environment they can decide everything by themselves. For anarchists workers' self-management is the thing that makes the difference. Even world revolution, if that leads to a world-scale planned economy (command economy), is not enough for anarchists or supporters of workers' self-management. They think that capitalism can be overcome simply taking the means of production in our own hands.