Despite being particularly illuminating, the answer by Lawrence B. Crowell is a bit too technical regarding the general nature of the question.
As you guess in your question, superconductivity is a concept with many many different facets. To list a few of them : superconductivity is
- a state of matter characterised by zero resistance flow of electric current and perfect diamagnetism. That's the original observation by Kammerling-Onnes and Meissner & Ochsenfeld in the 10's - 20's of the XX-th century.
- almost a Bose-Einstein condensate of charged particle. This is how London understood superconductivity, leading to the modification of the Maxwell equations in superconductors, and the first description of the superconducting phenomenology: the penetration effect and the notion of fluxoid in non-singly-connected superconductors, see e.g. the book by London [London, F. (1961). Superfluids, volume I: Macroscopic theory of superconductivity. book, Dover Publications, Inc.]
- a second order phase transition driven by a spontaneously symmetry breaking. In particular, the order parameter corresponds to the gap which opens when passing from a gapless metallic phase to a superconducting phase. A model describing the gap effects and useful for drawing simple pictures of the superconducting phenomenology is called the semi-condutor picture of superconductivity.
- an emerging macroscopic quantum phenomena. This can be seen in many ways: the simplest one being that the order parameter is a complex function, i.e. it can be seen as a macroscopic wave function, see e.g. the Ginzburg-Landau formalism
- a phase of matter with phase rigidity and long range order (Note: phase of matter refers to the state of matter, whereas phase rigidity refers to the phase of the macroscopic wave function of the order parameter.) The phase rigidity explains the persistance of current over macroscopic distances, the Josephson effect, the presence of vortex, ... Say quickly it does not give much more insight than the Ginzburg-Landau formalism, but it has a nice name :-)
- a phase with topological defects in the form of vortex and lattice of vortices. This is the famous Abrikosov lattice, historically found from the Ginzburg-Landau formalism. Please do not confound topological defect and topological phase, though for superconductors these two notions are intricate, see below. For a general overview of topological defects, I suggest the review by [Mermin, N. D. (1979). The topological theory of defects in ordered media. Reviews of Modern Physics, 51, 591–648.] unfortunately behind paywall.
- a transmutation phenomena from fermionic electrons to Cooper pairs (in the ground state) and quasi-particles (as excitations). This is the basic idea behind the Bardeen-Cooper-Schrieffer formalism, in addition to the phase transition. At the heart of the BCS formalism lies the guess of a microscopic Ansatz for the wave function of a superconductor in its ground-state (at zero-temperature if you prefer), now called the BCS Ansatz. This Ansatz is nothing but the coherent state one can construct from a Fermi sea (a gas of electrons at zero temperature if you prefer), and it associates pairs of electrons into a pseudo-bosonic state, though the exact nature of a Cooper pair is not really bosonic. So in short the state at zero temparature of a metallic phase is not the usual Fermi sea, but a quantum gas of BCS correlations (perhaps better to say a quantum liquid in that case) when there is attraction between electrons mediated by phonons, the so-called Cooper mechanism. The BCS formalism (especially the reconstruction of BCS ideas by Gor'kov using mathematical tools inherited from quantum-field theory) describes the phase transition, describes the famous isotope effect (demonstrating how the phonons participate to the onset of superconductivity), and allows to demonstrate the validity of the Ginzburg-Landau approach, so it describes as well all the phenomenologies already mentionned above. There is still some debates whether the BCS formalism describes the London momentum effect, but that's an other story full of polemics, so it has nothing to do in this overview.
- an example of a Higgs phenomena. I will not discuss this much, since it is almost completely encoded in the Ginzburg-Landau theory, at least at the basic level I would like to keep this discussion.
- a topological phase of matter (and here comes the beast !)
and I do not speak of low-dimensional superconductors which you mention in your question, e.g. phase of vortex-antivortex and the Kosterlitz-Thouless phase transition, or the Josephson effects, or the competition between superconducting and magnetic orders, ... just to focus on the general concepts.
I've tried to adopt a historical classification above, for the sake of convenience. Surely this list is not complete, since we do not understand completely the phenomenon associated with superconductivity (as in high-temperature superconductors, or iron-based superconductors for instance).
One has to note that the different facets listed above describe some of the phenomenologies of superconductivity, but most of them do not describe all the phenomenology. Clearly, the last sentence is tautological, since it's almost clear we do not know yet the complete phenomenology of superconductivity. I've tried to list quickly the phenomenologies described by all approaches.
There is not a lot of documentation about the topological aspects of superconductivity, since it's merely a simple rephrasing of the phenomenology in a new mathematical language. Using the terminology and the concepts of topological phase, one can describe the response of a superconductor under magnetic field, especially the generation of vortices. To the best of my knowledge the Josephson effect and the London momentum effect are not yet described in term of topological state of matter. For more details about the topological phase concepts behind superconductivity you can check the article by
Hansson, T. H., Oganesyan, V., & Sondhi, S. L. (2004). Superconductors are topologically ordered. Annals of Physics, 313, 497–538, also on arXiv:0404327 (open access).
In this paper, the first section is aimed as an introduction to the concepts of topological phases, which the authors demonstrate in the following sections. I think the first section is easy to follow, so I stop my answer here, and I let you come back with more specific questions if you still have some.
Have fun !
Post-scriptum: What Lawrence B. Crowell partly tried to explain (the answer is far more richer) is the so-called Anderson's theorem : as long as there is no time-reversal symmetry breaking interaction in the superconductor (like e.g. magnetic impurities or magnetic field), the superconducting phase is robust. This is one example of a symmetry protected topological state then.