If you want to apply for a Czech long-term visa, get a Trade licence or register a company, among other documents you should provide a criminal clearance report from your home country. If you are applying for a long-term visa you should also provide criminal clearance report from the country/countries in which you have resided continuously for more than 6 months in the last 3 years, or, in the absence of such a document, the applicant’s statement (Only for US citizen) Certificates from criminal records are issued by the local police department in your home country. It should be officially translated to the Czech language and it might need an apostille or super legalization.
If you are US citizen you should make an appointment at the US embassy in the Czech Republic and sign an affidavit confirming you haven’t committed any crime. You can make an online appointment at the US embassy, they will provide the proper form to fill out and sign in person. The cost of an affidavit is 50 US $.