In the United States, there is a procedure to get a passport without a birth certificate by obtaining sworn affidavits from two persons who have known you all your life to prove your identity and citizenship. I do not know what the State Department calls those sworn affidavits, but you could always search the US State Department's web site to find out.
Edit: I found it on a PDF provided by the State Department in the section titled Proof of Identity. It states:
"IF YOU CANNOT PROVIDE DOCUMENTARY EVIDENCE OF IDENTITY as stated above, you must appear with an IDENTIFYING WITNESS, who is a U.S. citizen, non-citizen U.S. national, or permanent resident alien that has known you for at least two years. Your witness must prove his or her identity and complete and sign an Affidavit of Identifying Witness (Form DS-71) before the acceptance agent. You must also submit some identification of your own." This proves who you are (identity).
So, apparently the conditions have changed since I was a kid getting my first passport.
In addition to this, you have to prove that you are a citizen by birth. That is covered in another section titled Proof of U.S. Citizenship, where on the last line it states "Notarized affidavits (or DS-10, Birth Affidavit) of older blood relatives having knowledge of your birth may be submitted in addition to some of the records listed above."
