Prove your connection to the country of passport application. To be considered eligible to hold a passport of any nation, you must have the right to reside in the country. There are six "connections" that could grant you a right to reside. These are marriage, connection to relatives, employment, retirement, as a refugee or for investment. The criteria regarding each connection varies for every country. You must consult the qualifications specific to the country.
Qualify for residency. You cannot hold a passport from any country unless you are qualified as a citizen. Therefore, you must obtain dual citizenship through one of the connections listed in Step 1. Each country holds different criteria for how to become a naturalized citizen. You must consult your country for specific immigration guidelines to determine what must be done. Most countries require a period of residency of up to 12 years before citizenship is granted. In cases of direct parentage, such as if you are the child of an American and a Canadian, you are automatically considered eligible for dual citizenship of both countries. To receive citizenship of the country you believe you have a connection to, apply directly through the country's department of immigration. The application process and paperwork is specific for each connection.
Apply for a passport with your second citizenship. Once you have been granted citizenship, you can apply for the second passport through the nation of citizenship. The process is lengthy unless you are already a dual citizen, though the rewards of travel, tax exemptions and work are worth the time and effort.