After reading the question Mentor trying to be first author?, I got the impression that (at least in the writers' field) the "corresponding author" flag could carry an implied meaning, such as "did more work than the other authors", or "is a more senior author". Is it the case?
I am in mathematics, a field where we typically sort the authors alphabetically, and I've never given any importance to that role (apart from "this author is willing to answer a couple of extra e-mails, if needed").
Corresponding authorship is often used to signal the person who is the primary contributor to the work. I've seen this in fields like economics where there is a relatively strong norm of listing authors alphabetically. In these cases, the primary contributor is signaled with the corresponding author designation. I've seen people go so far as to list corresponding authorship on their CVs when they list their papers although I think this is far from the norm.
In fields where authorship is not simply alphabetic, I've seen the corresponding author designation used to flag the second author as a major contributor because, even in cases of so called "dual-first authorship," somebody's name has to come first.
In lots of other fields, it caries little signal value and really is just used to mark the person who is in the best position to answer questions about the work. Of course, the person marked as best able to answer question will, mostly likely, be someone who was involved in a wide variety of the research reported in the paper.
In theoretical computer science, "corresponding author" does not carry any implied meaning related to who is the primary contributor.
The coauthors are listed in alphabetical order. The corresponding author might be, for example, the coauthor who took care of the interaction with the publisher, or a coauthor with a fairly permanent mailing address.
In pre-prints, conference abstracts, etc. we do not usually mention any corresponding author. This is just a piece of information that is required by journal publishers.
There is no prestige related to it. Certainly you will not list this kind of information in your CV.
I am sure it can be used for many meanings. I will provide the ones with which I am familiar:
First, corresponding author means it is the person to contact if you have questions about the research. The reasons for this is what varies.
Sometimes you will see first authors stating they are corresponding author. This is mostly implied by default in fields where the first authorship means also doing most of the intellectual work.
I am sure there are additional examples which hopefully are added by additional posts.
The Max-Planck-Gesellschaft, a German research funder, uses corresponding authorship as a criterion for paying publication costs with open-access publishers (source).
In this context, the corresponding author is the one whose funder paid for the publication.
In my field of Physics, there seems to be no implied meaning - the 3 papers that I have got published recently, I have been the first and corresponding author, primarily due to, as Jukka and Peter have mentioned, to doing most of the legwork and having the responsibility for the research.
Also, it can be seen as a logisitcs thing, where the corresponding author is the port-of-call during the proofing/editing stages by the journal.
In biology, the designation "corresponding author" is sometimes used to indicate the senior author(s) of a paper. A senior author is typically the professor, which may have a central role in the project but will usually not do any of the actual experimental work. Usually the senior author is listed last, but in cases of collaborations where there are multiple senior authors that contributed equally, this would sometimes be indicated by giving these authors a "corresponding author" designation.
However even within biology there is a lot of variation and this could mean different things.
The corresponding author is the author designated to be the first point of contact by readers of the paper. In theory, authors could use any rule they like to decide which author is the corresponding author.
However, in my experience, the corresponding author is often the author who has made the largest contribution or an author who has made one of the larger contributions to the paper. The corresponding author is also often chosen as someone who is likely to have a stable contact address and is likely to remain in academia for some time to come. In that case, supervisors are sometimes corresponding authors rather than students (even where the student has done most of the work). A similar pattern can hold in the case of industry collaborations where the academic may be the corresponding author.
Importantly, in most models of ethical publishing, all authors are meant to take responsibility for the claims in the published work.
