In "commutative geometry," I think bimodules tend to be a little concealed. People are more likely to talk about "correspondences" which are the space version of bimodules: A correspondence between spaces X and Y is a space Z with maps to X and Y.
When you think in this language, there are lots of examples you're missing. For example, the right notion of a morphism between two symplectic manifolds is a Lagrangian subvariety of their product, or even a manifold mapping to their product with Lagrangian image (maybe not embedded). See, for example, Wehrheim and Woodward's Functoriality for Lagrangian correspondences in Floer homology.
Similarly, correspondences are incredibly important in geometric representation theory. See, for example, the work of Nakajima on quiver varieties.
The theory of stacks also is at least partially founded on taking correspondences seriously as objects, and in particular being able to quotients by any (flat) correspondence.
This same philosophy also underlies groupoidification as studied by the Baez school (they tend to use the word "span" instead of "correspondence" but it's the same thing).
I know almost nothing about noncommutative rings, but I have thought a bit about what the general concept of spectra might or should be, so I'll venture an answer.
One other property you might ask for is that it has a good categorical description. I'll explain what I mean.
The spectrum of a commutative ring can be described as follows. (I'll just describe its underlying set, not its topology or structure sheaf.) We have the category CRing of commutative rings, and the full subcategory Field of fields. Given a commutative ring $A$, we get a new category $A/$Field: an object is a field $k$ together with a homomorphism $A \to k$, and a morphism is a commutative triangle. The set of connected-components of this category $A/$Field is $\mathrm{Spec} A$.
There's a conceptual story here. Suppose we think instead about algebraic topology. Topologists (except "general" or "point-set" topologists) are keen on looking at spaces from the point of view of Euclidean space. For example, a basic thought of homotopy theory is that you probe a space by looking at the paths in it, i.e. the maps from $[0, 1]$ to it. We have the category Top of all topological spaces, and the subcategory Δ consisting of the standard topological simplices $\Delta^n$ and the various face and degeneracy maps between them. For each topological space $A$ we get a new category Δ$/A$, in which an object is a simplex in $A$ (that is, an object $\Delta^n$ of Δ together with a map $\Delta^n \to A$) and a morphism is a commutative triangle. This new category is basically the singular simplicial set of $A$, lightly disguised.
There are some differences between the two situations: the directions have been reversed (for the usual algebra/geometry duality reasons), and in the topological case, taking the set of connected-components of the category wouldn't be a vastly interesting thing to do. But the point is this: in the topological case, the category Δ$/A$ encapsulates
how $A$ looks from the point of view of simplices.
In the algebraic case, the category $A/$Field encapsulates
how $A$ looks from the point of view of fields.
$\mathrm{Spec} A$ is the set of connected-components of this category, and so gives partial information about how $A$ looks from the point of view of fields.
Best Answer
In accordance with the suggestion of Yemon Choi, I am going to suggest some further delineation of the approaches to "Non-commutative Algebraic Geometry". I know very little about "Non-commutative Differential Geometry", or what often falls under the heading "à la Connes". This will be completely underrepresented in this summary. For that I trust Yemon's summary to be satisfactory. (edit by YC: BB is kind to say this, but my attempted summary is woefully incomplete and may be inaccurate in details; I would encourage anyone reading to investigate further, keeping in mind that the NCG philosophy and toolkit in analysis did not originate and does not end with Connes.)
Also note that much of what I know about these approaches comes from two sources:
The paper by Mahanta
My advisor A. Rosenberg.
Additionally, much useful discussion took place at Kevin Lin's question (as Ilya stated in his answer).
I think a better break down for the NCAG side would be:
A. Rosenberg/Gabriel/Kontsevich approach
Following the philosophy of Grothendieck: "to do geometry, one needs only the category of quasi-coherent sheaves on the would-be space" (edit by KL: Where does this quote come from?)
In the famous dissertation of Gabriel, he introduced the injective spectrum of an abelian category, and then reconstructed the commutative noetherian scheme, which is a starting point of noncommutative algebraic geometry. Later, A. Rosenberg introduced the left spectrum of a noncommutative ring as an analogue of the prime spectrum in commutative algebraic geometry, and generalized it to any abelian category. He used one of the spectra to reconstruct any quasi-separated (not necessarily quasi-compact), commutative scheme. (Gabriel-Rosenberg reconstruction theorem.)
In addition, Rosenberg has described the NC-localization (first observed also by Gabriel) which has been used by him and Kontsevich to build NC analogs of more classical spaces (like the NC Grassmannian) and more generally, noncommutative stacks. Rosenberg has also developed the homological algebra associated to these 'spaces'. Applications of this approach include representation theory (D-module theory in particular), quantum algebra, and physics.
References in this area are best found through the MPIM Preprint Series, and a large collection is linked here. Additionally, a book is being written by Rosenberg and Kontsevich furthering the work of their previous paper. Some applications of these methods are used here, here, here, and here. The first two are focusing on representation theory, the second two on non-commutative localization.
Kontsevich/Soibelman approach
They might refer to their approach as "formal deformation theory", and quoting directly from their book
Their approach is related to $A_{\infty}$ algebras and homological mirror symmetry. References that might help are the papers of Soibelman. Also, I think this is related to the question here. (Note: I know hardly anything beyond that this approach exists. If you know more, feel free to edit this answer! Thanks for your understanding!)
(Some comments by KL: I am not sure whether it is appropriate to include Kontsevich-Soibelman's deformation theory here. This kind of deformation theory is a very general thing, which intersects some of the "noncommutative algebraic geometry" described here, but I think that it is neither a subset nor a superset thereof. In any case, I've asked some questions related to this on MO in the past, see [this][22] and [this][23].
However, there is the approach of noncommutative geometry via categories, as elucidated in, for instance, [Katzarkov-Kontsevich-Pantev][24]. Here the idea is to think of a category as a category of sheaves on a (hypothetical) non-commutative space. The basic "non-commutative spaces" that we should have in mind are the "Spec" of a (not necessarily commutative) associative algebra, or dg associative algebra, or A-infinity algebra. Such a "space" is an "affine non-commutative scheme". The appropriate category is then the category of modules over such an algebra. Definitively commutative spaces, for instance quasi-projective schemes, are affine non-commutative schemes in this sense: It is a theorem of van den Bergh and Bondal that the derived category of quasicoherent sheaves on a quasi-projective scheme is equivalent to a category of modules over a dg algebra. (I should note that in my world everything is over the complex field; I have no idea what happens over more general fields.) Lots of other categories are or should be affine non-commutative in this sense: [Matrix factorization categories][25] (see in particular [Dyckerhoff][26]), and probably various kinds of Fukaya categories are conjectured to be so as well.
Anyway I have no idea how this kind of "noncommutative algebraic geometry" interacts with the other kinds explained here, and would really like to hear about it if anybody knows.)
Lieven Le Bruyn's approach
As I know nearly nothing about this approach and the author is a visitor to this site himself, I wouldn't dare attempt to summarize this work.
As mentioned in a comment, his website contains a plethora of links related to non-commutative geometry. I recommend you check it out yourself.
Approach of Artin, Van den Berg school
Artin and Schelter gave a regularity condition on algebras to serve as the algebras of functions on non-commutative schemes. They arise from abstract triples which are understood for commutative algebraic geometry. (Again edits are welcome!)
Here is a nice report on Interactions between noncommutative algebra and algebraic geometry. There are several people who are very active in this field: Michel Van den Berg, James Zhang, Paul Smith, Toby Stafford, I. Gordon, A. Yekutieli. There is also a very nice page of Paul Smith: noncommutative geometry and noncommutative algebra, where you can find almost all the people who are currently working in the noncommutative world.
References: [This][16] paper introduced the need for the regularity condition and showed the usefulness. Again I defer to [Mahanta][17] for details. Serre's FAC is the starting point of noncommutative projective geometry. But the real framework is built by Artin and James Zhang in their famous paper [Noncommutative Projective scheme][18].
Non-commutative Deformation Theory by Laudal
Olav Laudal has approached NCAG using NC-deformation theory. He also applies his method to invariant theory and moduli theory. (Please edit!)
References are on his page [here][19] and [this][20] paper seems to be a introductory article.
Apologies
Without a doubt, I have made several errors, given bias, offended the authors, and embarrassed myself in this post. Please don't hold this against me, just edit/comment on this post until it is satisfactory. As it was said before, the [nlab][21] article on noncommutative geometry is great, you should defer to it rather than this post.
Thanks!
[16]: https://books.google.com/books?hl=en&lr=&id=_BnSoQSKnNUC&oi=fnd&pg=PA33&dq=%252522Artin%252522+%252522Some+algebras+associated+to+automorphisms+of+elliptic+curves%252522+&ots=hRXnP7udMW&sig=t77CnWnsYPHhuonQQffrSXedyj0#v=onepage&q="Artin" "Some algebras associated to automorphisms of elliptic curves"&f=false [17]: https://arxiv.org/abs/math/0501166 [18]: https://web.archive.org/web/20121023193142/http://www.ingentaconnect.com/content/ap/ai/1994/00000109/00000002/art01087 [19]: https://web.archive.org/web/20181103123848/http://folk.uio.no:80/arnfinnl/ [20]: https://web.archive.org/web/20080425144650/http://folk.uio.no/arnfinnl/Noncom.alg.geom.pdf [21]: https://ncatlab.org/nlab/show/noncommutative%20geometry [22]: What is a deformation of a category? [23]: Deformation theory and differential graded Lie algebras [24]: https://arxiv.org/abs/0806.0107 [25]: Matrix factorizations and physics [26]: https://arxiv.org/abs/0904.4713