Universality, or the concept of "universal," is extremely wide-ranging in Hegel, but in general it denotes that which transcends the subjective and the particular. The nature and essence of Spirit in and of itself is universal, but universality is only one aspect of Spirit as it unfolds in the world. The opposite aspect is particularity, and the division between it and the aspect of subjectivity is based on the division Spirit creates within itself as it becomes self-conscious (which involves the knowing of itself as an object rather than just a subject).

The course of history is driven by the dialectic (the back-and-forth) between the universal and particular aspects of Spirit. These aspects are sometimes joined, when the State succeeds in unifying the particular and subjective wills of its citizens with the universal principle that is the common Spirit of the people. 

Universality, whether it is fully meshed with the particularities of culture or not, must be present in a culture before that culture can be considered a State (since the State is the practical embodiment of a universal national principle). Until this happens, true "history" has not begun for that culture. Universality is first introduced in a culture by thought, which rejects traditional, unconsidered ideas of duty in favor of universal, rational laws. Thus, human culture seeks to know itself in a universal context, even as Spirit seeks to know itself as an objective thing in the world.

Popular pages: Introduction to the Philosophy of History