Spectra of $C^*$ algebras

The spectrum of $L^\infty(R)$ is the hyperstonean space associated with the measurable space R. More information can be found in Takesaki's Theory of Operator Algebras I, Chapter III, Section 1.

For $L^\infty(X)$, the spectrum is the Stone space of the algebra of measurable sets mod null sets. This is because a character is determined by what it does on characteristic functions because their span is dense.

Your question (especially the first part) is a bit vague, but I'll shoot: A very nice example is provided by Carleson's corona theorem, stating that the unit disk is dense in the spectrum of the Hardy space $H^\infty$ (the bounded holomorphic functions on the unit disk).

As for the spectra of $L^\infty$, I don't think you can ever come up with a concrete example of a character on this space. You actually need the axiom of choice to prove that the spectrum is nonempty. Likewise with the points of the spectrum of $H^\infty$ outside the unit disk.