What is an interpretation of quantum mechanics? In the sense of "Copenhagen Interpretation", what exactly is an interpretation?
What purpose does an interpretation serve? Can an interpretation be tested or even be correct or incorrect independent of the underlying theory being interpreted? Why do physicists publish them?
This topic is currently dominating another question of mine:
Why does the Copenhagen interpretation assert randomness if this cannot be tested?
This question is a proper place for those arguments.
This is not a subjective question. Please, no opinions. What do the authors of these interpretations understand them to be? Try to cite sources.
 A: 
In the sense of "Copenhagen Interpretation", what exactly is an interpretation? What purpose does an interpretation serve?

I would describe interpretations of quantum mechanics as part of the philosophy of physics. Here is a well-known quote by Bertrand Russell: "As soon as definite knowledge concerning any subject becomes possible, this subject ceases to be called philosophy, and becomes a separate science." This mathoverflow answer and the comments on it may also be helpful. The Copenhagen interpretation (CI) was the first interpretation of quantum mechanics, and it arose at a time when there was honest confusion and doubt about what quantum mechanics meant. The "shut up and calculate" option was not necessarily available, because the rules for calculation weren't yet clear. The CI was developed around 1925-1927. Here are a couple of examples of developments that occurred during that period, which show how unclear the situation was. The Heisenberg uncertainty principle dates to 1927 (and was not yet properly understood when Heisenberg first published it). Also, Bohr had been pushing the idea that quantum mechanics applied to matter but not to light (the Bohr-Kramers-Slater theory), and this was disproved experimentally by Bothe and Geiger in 1925. So the general idea is that this was a period when people were struggling to achieve the kind of "definite knowledge" referred to by Hilbert. After 1927, they were well on the road to "definite knowledge," and the philosophy of quantum mechanics had made itself nearly obsolete -- which, in in Hilbert's description, should be the goal of every sub-field of philosophy. One way we can tell that this process of self-obsolescing was completed after not much longer is that in the postwar period, American textbooks almost all omitted any discussion of interpretations.[Osnaghi 2009]

Can an interpretation be tested or even be correct or incorrect independent of the underlying theory being interpreted? 

The part about "independent of the underlying theory being interpreted" is important. It restricts us to talking about testing one interpretation against another, as opposed to testing quantum mechanics in general. The Copenhagen interpretation's main competition is the many-worlds interpretation (MWI), which came much later, in 1957. WP has a bunch of material on this topic, which I think gives a fair picture of the scientific consensus.

MWI is considered by some to be unfalsifiable and hence unscientific because the multiple parallel universes are non-communicating, in the sense that no information can be passed between them. Others[53] claim MWI is directly testable. Everett regarded MWI as falsifiable since any test that falsifies conventional quantum theory would also falsify MWI.[19] 

(The second reference is to [Everett 1957].)
If you look through this whole section of the WP article, I think you'll see that there's an overwhelming consensus that interpretations are not testable in the sense of testing one interpretation versus another. As suggested by the quote above, they are only testable in the sense that one could try to falsify quantum mechanics in general.
I think MWI was a positive contribution because before it came along, there was only CI, and one could have gotten the impression that CI was the only possible correct interpretation of quantum mechanics. (Many people still have this impression.) MWI was a good antidote to this kind of smugness. It showed that when we talk about what quantum mechanics means, there is the potential for a huge amount of ambiguity, and this ambiguity is never going to be settled. That is, according to our current understanding of the standard model, there is no experiment, even in principle, that could ever decide between CI and MWI. Therefore we can tell that any question that is answered differently by CI and MWI is a question that can never be answered empirically. For instance, one can never determine empirically whether quantum processes are "really" random as in CI, or whether the evolution of physical states "really" goes through non-unitary steps. A discussion of the complicated historical context is given in [Osnaghi 2009].
Another salutary effect of MWI was that it raised the issue of what "measurement" was in quantum mechanics. For example, Stern, describing Bohr's view of Everett's work, said, "[...] the basic shortcoming in his method of
approach [...] is his lack of an adequate understanding of the
measuring process." This has to some extent been clarified in terms of decoherence.[Zurek 2001]
References
Everett,  "Relative State Formulation of Quantum Mechanics," Reviews of Modern Physics 29 (1957) 454–462. Everett's thesis on this topic is available online: http://www.pbs.org/wgbh/nova/manyworlds/pdf/dissertation.pdf
Osnaghi, Freitas, and Freire, "The Origin of the Everettian Heresy," Studies in History and Philosophy of Modern Physics 40 (2009) 97–123, http://stefano.osnaghi.free.fr/Everett.pdf
Zurek, "Decoherence, einselection, and the quantum origins of the classical," 2001, http://arxiv.org/abs/quant-ph/0105127
A: [Borrowed disclaimer: this is a personal view. It is not necessarily reflective of what physicists at large think, nor is it accepted canon]
The reason is that taking the metaphysical part out of physics takes most of the "fun" out of it. People become physicists for different reasons, but many, like me, started thinking we could ultimately learn something about the fabric of reality. If you are only left with measurements, you are an excellent technician, but not an actual scientist, at least not in the spirit of the "founding fathers". Physics without some metaphysics becomes engineering (I am not trying to be disrespectful of engineering).
Articles with a heavier component of interpretation are perhaps published on the verge or after a change of paradigm, then mostly relegated to thesis, fringe journals, popular science articles, or books. But they are still out there and coming in droves. And I love them if they are good. 
A: Personally, I think the primary reason for thinking about interpretations is that they can lead to new predictions. They generally don't do this directly - almost by definition, an interpretation makes no new predictions by itself - but by changing the landscape within which new theories can be proposed.
The best example I know of is the Lorentz equations for time dilation and length contraction. These were known before Einstein of course, but they lacked an interpretation that made sense. Einstein provided that interpretation, by changing our understanding of space and time. Without Einstein's interpretation, there's no reason why we couldn't have kept on using the Lorentz equations to make all the same predictions that special relativity makes - but there's no way we could have gone from there to general relativity. By changing the interpretation you don't change the predictions a theory makes, but instead you change the ways in which the theory can be meaningfully altered.
It seems to some researchers that quantum mechanics is in a similar state to relativity pre-Einstein. We have the equations and understand very well how to use them. But at the same time, the theory doesn't quite make sense. It hasn't been unified with general relativity, and we don't know how to change it in such a way that it can be. (There is a lot of research pushing at the interface between the two theories, but something fundamental still seems to be missing.) With a different interpretation of quantum mechanics, we might see the way to do this. I think the possibility of doing for quantum mechanics what Einstein did for the Lorentz equations is what motivates at least some researchers to think about quantum interpretations.
A: Simple and to the point (some people might refer to this as KISS principle).
Assuming that theory refers to a mathematical theory, then the
interpretation is that thing that tells when and where to apply the theory
Simple as that, no more no less.
UPDATE (after comment and downvote):
"In a more general context, that would be a reasonable way to define "interpretation." But in this context it's wrong. Interpretations of quantum mechanics do not tell us when and where to apply quantum mechanics. If they did, then they would have experimentally testable implications -- which they don't." –  Ben Crowell
The direct part. The fact that the interpretation of the ontology and range of application of QM is restricted to atomic scales is a testimony to this. 
But the theory can be used both to economics (e.g martingales),  condensed matter physics, biology et al. The interpretation of QM (and the ontology) restricts the application of the theory there (and any testable implications which would require a re-interpretation).
Now the converse part. The restriction of the range of apllication of the QM to atomic only scales, necesarily has ontological implications (which further provide a "rationalisation" of the restriction). The application to these areas (where mathematicaly the same formalism can be used), would change the ontology of the theory so as to accomodate the extensive range of application and would lead to testable or falsifiable implications and predictions. Why? simply because the (new re-interpreted) theory would provide results to be tested.
Note that, in the argument above, the mathematical formalism is not affected, only its range of application (as a result its "ontology", or what constitutes a concrete model of the theory).
