Emergence is a classical concept in systems theory, where it denotes the principle that the global properties defining higher order systems or "wholes" (e.g. boundaries, organization, control, ...) can in general not be reduced to the properties of the lower order subsystems or "parts". Such irreducible properties are called emergent.
Self-organization may be defined as a spontaneous (i.e. not steered or directed by an external system) process of organization, i.e. of the development of an organized structure. The spontaneous creation of an "organized whole" out of a "disordered" collection of interacting parts, as witnessed in self-organizing systems in physics, chemistry, biology, sociology ..., is a basic part of dynamical emergence.
So, now I'm thinking of writing a sci-fi novel where the revolution is referred to as The Emergence. It's even better than the Singularity!
But anyway, the second definition is particularly useful. If we see the capitalist system as a whole, and do not conceptualise the working class as a discrete, identifiable (i.e. ordered) element within this system, we can see the development of a socialist society as a process of organisation.
In this process, the working class moves from a "'disordered' collection of interacting parts" which share basic characteristics, to a highly ordered group. This group then engulfs particular elements of the system as a whole while discarding or destroying others, so that we can say that a new society emerges.
The role of the revolutionary organisation then becomes guiding this process of self-organisation in such a way that the new society can emerge.