How to cite authors from any country correctly? It has always seemed to me that the Mathematical Community gives a high importance to the act of properly citing an author (Do not write Erdos! It's Erdős. Cauchy must be read as in French, not as in English...).
Hence, I thought that it might be useful to ask your opinion about how to cite correctly a foreign and/or complicated author name in the references.
It seems that Bibtex handles only standard English style names well, and requires many workarounds for foreign names (see here). So it is better not to count on it too much.
Feel free to point out any suggestion, or even the problems you have come across when citing authors.
I'll start a list:


*

*✔ First, Middle, Last - Names. 
OK, I think there are no doubts, "John Horton Conway" should be cited as "J. H. Conway". Note the whitespaces after each period.

*✔ Spanish names: See the very good answer of Leo Alonso.

*✔ "Nobiliary" names (von, van der, etc.): See the answer of 
R. van Dobben de Bruyn.

*Chinese names: Here I am often in trouble. I read that Chinese write their surname first and then their given name, but I think that in papers they are usually swapped. Also I heard that many Chinese have the same surname.
THE FREE LOOKUP ANSWERS


*

*The user zeno pointed out that the tool MRLookUp http://www.ams.org/mrlookup is free an can be used to get the bibtex of any article tracked by the AMS.

*Federico Poloni also suggested MRef http://www.ams.org/mref.
THE "LET'S JUST CLOSE" ANSWERS
Some users voted to close this question for different reasons. I answered this question thinking that how to cite an author correctly is often a problem for mathematicians, so address it on MO could have been useful for many users. I can agreed to close this question, probably choosing zeno answer as the best, BUT before I would like to see more comments and opinions, especially about Asian names. 
NOTES:
1) Many answered: "just look on MathSciNet". Unfortunately, MathSciNet is not free for everyone, so I think this is a quite unsatisfactory method.
2) The question is not about the transliteration of names, you can assume that the author already have a name written in a reasonable set of characters extending the Latin. The question is about the abbreviation of names in references.  
 A: Two somewhat similar points:
1)  When it comes to choosing a correct spelling, I disagree with Alexandre Eremenko that one should follow the spelling in the cited paper.  I think one should always use the spelling under which the person is best known.  
Let's say you are citing several papers by Erdős, some with a non-standard spelling (MathSciNet also lists Erdös, Erdós, Erdőš, etc.)  Are you going to have several different spelling of the same author?  Only in the references or in the main body of the paper as well?  This only creates a confusion and serves no clear purpose (MSN merges all spellings anyway).  
Some Russian names gives more unfortunate examples. Are you going to cite Oleinik or Oleĭnik?  Gelfand or Gel'fand?  The latter spelling is standard in both cases, but few non-Russian speakers would know how to pronounce the names in that case - these extra symbols again create an unnecessary confusion.
2)  For Chinese and other foreign names - again, the golden rule is to use the name they are best known.  I recommend reading this Wikipedia guide which I found to be well thought out and quite helpful.  
A: There is also the singular case of Yahya ould Hamidoune, the Mauritanian mathematician. In fact, ould is not a name: it just means 'the son of'. It is used in the same spirit of the Italian patronyms de or de' (as in Bruno de Finetti or Lorenzo de' Medici), so it should be treated in the very same way. However, most of Hamidoune's papers are cited (or even signed by Hamidoune himself!) as if 'Yahya' and 'ould' were two first names, which is why you will typically read 'H. O. Hamidoune' or 'Hamidoune, H. O.' in bibliographies. (In still other cases, matters are just cut short by dropping the 'ould'.)
In an ideal world, I think the mistake (if you also take it as such) should be definitely fixed: Hamidoune's papers would be indexed as 'H. ould Hamidoune' or 'ould Hamidoune, H.' (depending on the journal style), and ordered by making reference only to 'Hamidoune' (if they have to be ordered alphabetically). But I'm urged to be realistic: this won't happen for many reasons.
A: MathSciNet is not free, but MRLookUp 
http://www.ams.org/mrlookup
is free. So just look it up there and copy the bibtex.
A: For spaniards names.
A spaniard has a given name that sometimes is two (even more) like Pedro, Pablo, Juan Luis, María Eugenia, etc. The second given name is recognized because it is usually different from the usual family names. It is possible to have a family names like Carlos, but it is not quite frequent.
Additionally all the people has two family names, the first comes from the father's family and the second from mother's the family. Roughly this second family name works as the "middle initial" in some English speaking countries.
So, a mathematician named José Luis García Pérez should be cited like
García Pérez, José L.

In real life you might call him José, José Luis, or perhaps Pepe.
A really brief way and of course correct of referring to him is
García, J. L.

He might be tired of explaining this subtleties to editors and authors and write his name as José Luis García-Pérez, avoiding references to him as prof. Pérez instead of the correct prof. García. But this solution is not really compatible with the legal use stablished in Spain.
It happens that some people have complicated family names like Francisco Javier de la Vega Martínez, then the citation should be
Vega Martínez, Francisco J. de la

and the short version
Vega, F. de la

Another source of possible confusion is people with composite last names like, say Pedro Antonio García-Valcárcel Rodríguez. Then the composite works a single last name and most of the time its is not abbreviated. Perhaps in a desperate situation one might out an abbreviation for the first of the two because it is usually the more common. A citation should read
García-Valcárcel Rodríguez, Pedro A.

and the (extra) short version
G.-Valcárcel, P.

I think I am covering the several cases. Now for the extremely complicated example in the question: Juan Pablo Fernández de Calderón García-Iglesias. The name "Juan" is the first given name, thus the main one, "Pablo" is the second given name, "Fernández de Calderón" is the first last name, so the main one and "García-Iglesias" is a composite second last name. The right way of citation:
Fernández de Calderón García-Iglesias, Juan P.

a short version
Fernández de Calderón, J.

And if you are really out of space the following
Fdez. de Calderón, J.

would be possible. There are other variants but I think I've covered the key points.
A: One thing that some people don't know is that in some languages, nobiliary particles¹ are neither capitalised (unless they appear at the beginning of the name; i.e. when initials are omitted)² nor alphabetised. In Dutch, for example, van, de, van der, etc are ignored for alphabetisation purposes. In Flemish names, however, the particles are usually both alphabetised and capitalised.
Example. (Dutch) B. L. van der Waerden should be under W, not V. Another format³ would be Waerden, B. L. van der.
Example. (German) C. L. F. von Lindemann should be under L, not V.
Example. (French) P. de Fermat should be under F, not D.
Example. (Flemish) M. Van den Bergh is under V! Note also the capitalisation of Van, but not den. This is presumably because the V occurs at the beginning of the name.
However, it seems that sometimes theorems named after people with such names may or may not (presumably depending on the country) drop the particle altogether: e.g. Fermat's little theorem, Lindemann–Weierstraß theorem. In (modern) Dutch, you would never do this; e.g. De Bruijn–Erdős theorem (note that De is capitalised, since the initials are omitted).
Remark. I find it entertaining to see that some authors are aware of these rules and others aren't (or does it depend on their BibTeX setup?). Next time you try to find a name of this form in the references, take a look at how the author alphabetised it.
Conclusion. Every country has different conventions, and it's very confusing. However, the strategy of following what the author uses herself is a safe one.

¹Their name is somewhat misleading: they do not always indicate nobility (e.g. in the Netherlands they don't).
²I believe that in German, the particle is omitted altogether if the surname is not preceded by the initials.
³This is the Dutch format. I don't know the conventions of other countries.
A: EDIT. As Dirk pointed, when you cite a paper (written in Latin font) you always
write the author's name exactly as it is written in the paper.
When you cite a paper which is written in some other alphabet,
the following are two reasonable approaches to this that I know: 1. To follow Mathscinet.
2. If an author ever wrote a paper using Latin font, write his name as he did himself.
Of course these approaches give the same result in most cases, but the second is especially important when you cite old authors who are not in Mathscinet,
for example, Chebyshev. (There was a nice joke in one combinatorics book: Problem. How many ways are there to spell Tschebyschev in English?).
It happens that the same author (or the translator) wrote his name with different spellings in different papers in English. In this case, when you cite his paper, just follow the spelling in the paper that you cite. 
EDIT. Here is a little example which shows how good the Mathscinet author database is. American mathematician D. F. Shea once published a paper in a Ukrainian journal, in Russian, and this paper was later translated to English. The translator of the paper transliterated his name from Russian as "Shia". Nevertheless Mathscinet determined his identity correctly.
EDIT2. There is a remarkable book on the subject of Chebysheff's names (first and last), and many other related important subjects:
The Thread: A Mathematical Yarn Paperback 
by Philip J. Davis , 1989.
Strongly recommended.
However, the author is not a native Russian speaker, so he does not discuss such advanced topics as the one in @Fedor Petrov comment.
