Hegel's most important, and most influential concept was the idea of a dialectic, or a conflict, between opposite ideas. Hegel thought that the history of philosophy was driven, in a sense, by these conflict, which took the form of a thesis (an idea), an antithesis (the opposite idea), and a synthesis (a sort of reconciliation between the two.) Hegel thought this conflict was always ongoing, and our understanding of it was essential to how we view the world.
Another important concept, one echoed in a sense by positivists like Auguste Comte and many other nineteenth century thinkers, was that the world was moving inexorably (through the dialectic) to a time when reason and a scientific worldview would supersede religion as the dominant framework for understanding the world. Karl Marx was also influenced by this idea, as well as that of the dialectic, though he would reframe it in terms of material, or economic realities.
Finally, another idea of Hegel's that was very influential in the nineteenth century was the idea that the nation-state was the embodiment of Spirit, or God immanent in the world. This idea, which operated on a very high level of abstraction, was nevertheless influential among nineteenth century liberal nationalists, especially in Hegel's native Germany, which would not achieve unification until the 1870s.