Form N-600 is issued by U.S. Citizenship and Immigration Services (USCIS). This form enables certain people who became U.S. citizens automatically, through their U.S. citizen parents or grandparents, to apply for a certificate proving their citizenship. The technical terms for gaining citizenship in such a way are either "acquisition" (when someone is born outside the U.S. to U.S. citizen parents) or "derivation" (when the parent(s) of a child who has a green card become U.S. citizens, most likely through naturalization).
The exact law that might apply to you cannot be summarized here, because it depends on the year in which you were born and the terms of the laws in effect at that time. If you're not sure whether you have in fact acquired or derived citizenship, speak to an immigration attorney.
Note that N-600 is NOT the form that should be used by people -- in particular, U.S. lawful permanent residents or green card holders -- who are applying to naturalize. Gaining citizenship through naturalization is a separate process in which applicants must file Form N-400 and prove that they have spent the appropriate amount of time in the U.S. with a green card, have good moral character, can speak, read, and write English, can pass a test in U.S. civics, and so forth. See Steps to Become an American Citizen for an overview on that process.
Of course, you'll also need to prove your eligibility, by submitting supporting documents, as described in USCIS's instructions for this form, found via the USCIS page. Below, we provide a quick summary of the most likely documents you'll need to submit and why they're important forms of proof.
You will, in the end, need to make your own judgments about which documents will prove that you have a valid claim to U.S. citizenship.
For example, if you were claiming to have acquired citizenship because you were born to U.S. citizen parents between January 13, 1941 and December 23, 1952, you'd need to prove not only that both your parents were in fact U.S. citizens, but that one had a prior residence in the United States. In such a case it would be important to submit proof of both your parents' U.S. citizenship (such as their birth or naturalization certificates), proof of your relationship to them (your own birth certificate), and proof of one parent's residence in the U.S. (such as copies of a home mortgage, pay stubs from a U.S. employer, and so forth).
Because your claim to citizenship rests upon your parent(s), you will almost need to provide your birth certificate showing your parent(s) names along with your Form N-600.
It's even possible that one of your parents may never have claimed U.S. citizenship, having gained it through his or her own parent(s). You would, in that case, also need to provide your parent's birth certificate showing relationship to your grandparents.
Under some of the earlier laws on acquisition of citizenship, you need to prove that your father was married to your mother at the time of your birth in order to gain citizenship through your father. Alternately, you can show that your father "legitimated" you by a certain age.
If your parents were indeed married, you'd want to submit their marriage certificate. If you were legitimated, you'd need to provide proof that your father complied with the legitimation laws of your home country, such as perhaps signing an affidavit confirming his parental responsibility.
If the law in effect when you were born requires you to have been unmarried when your parents naturalized, then you should submit any of your marriage certificates, along with a certificate showing any instance in which your marriage was terminated due to divorce or death.
Some of the laws on derivation of citizenship require you to have obtained a green card by a certain age, or before your parents became U.S. citizens. In such a case, you'll want to include a copy of your green card or other proof of permanent resident status along with your Form N-600.
All supporting documents accompanying the N-600 form should be readable copies. Don't send originals -- you won't get them back.
If the documents are in a language other than English, those need to be translated, word for word, by a competent translator, into English.
The translator should include a signed statement on the bottom of the translated version saying: "I, [name] certify that I am competent to translate from [the language of the document] to English and that the above is a true and correct translation to the best of my knowledge and belief." Below that, the translator should not only add a signature, but his or her address, telephone number, and the date.
If you have any questions at all about whether you have acquired or derived U.S. citizenship, or what documents will help prove your case, consult an immigration lawyer. The process of getting a certificate of citizenship can be lengthy under the best of circumstances. If USCIS isn't convinced by your documentary proof, the decision will drag on even longer, as it sends you letters asking for more documents, then takes time to review those documents, and so on. Having a lawyer's help will ensure that you present the best possible forms of proof to begin with.