Forms N-550 and N-570 are simply technical names given by the United States government to the Certificate of Naturalization that it gives to certain new U.S. citizens. The certificate serves as proof that the person whose name and photo the form bears has obtained U.S. citizenship via the process known as naturalization.
What Is Naturalization?
Naturalization is the process by which U.S. lawful permanent residents (green card holders) may, after a certain number of years, apply to become U.S. citizens. The process involves filling out and submitting U.S. Citizenship and Immigration Services (USCIS) Form N-400, attending an interview, demonstrating the ability to speak, read, and write English, and passing a test on U.S. history and government.
Do You Need to Separately Apply for the Certificate of Naturalization?
No, people who become naturalized citizens will automatically be given a certificate of naturalization at their swearing-in ceremony. You will be scheduled for a swearing-in ceremony after the USCIS interview at which you are approved for U.S. citizenship.
You will, however, separately need to apply for a U.S. passport. This will be an important document to have if you travel outside the United States. It is also easier to carry around with you than the certificate (especially if you decide to frame the certificate). You should be given an opportunity to fill out an application for a U.S. passport at your swearing-in ceremony. If you miss that opportunity, passport applications are handled by the U.S. State Department.
By the way, if you decide to frame your certificate of naturalization, you might want to make a photocopy first -- if, that is, you plan to file visa petitions in order to help any of your family members immigrate. (Yes, copying the certificate is illegal for most purposes, but it's okay if you're lawfully trying to petition for a family member.)
Is a Certificate of U.S. Citizenship the Same Thing?
No, a certificate of U.S. citizenship is the document given to people who became citizens automatically, through their parents or grandparents, via either the legal process of acquisition or derivation of citizenship.