Theo's question made me wonder if there are other "noncommutative analogs" outside of operator algebras. Some noncommutative analogs from operator algebras include:

• A $C^\ast$-algebra is a noncommutative topological space (cf. the Gelfand transform).
• The multiplier algebra of a nonunital $C^\ast$-algebra is the noncommutative Stone-Cech compactification.
• A spectral triple is a noncommutative manifold (add some extra data to the spectral triple to get a noncommutative Riemannian manifold cf. arXiv:0810.2088).
• A von Neumann algebra is a noncommutative measure space.

Are there any other good examples? If you know more in operator algebras, that's great too.

EDIT: these algebras should be considered as various functions spaces for noncommutative spaces as per @Yemon's answer. I'm going to leave the above text as is unless there are requests for another edit.