Under the right circumstances, a person is allowed to become a citizen of both Canada and the United States, simultaneously. Many people enjoy the benefits of dual citizenship, allowing them to to travel back and forth freely, vote, and otherwise take advantage of the rights of citizens on both sides of the U.S.-Canada border.
Before thinking about citizenship, however, you must find out whether you qualify to immigrate to either the United States or Canada in the first place. (We're assuming you're already a citizen of one of these countries.) Citizenship is the highest immigration benefit you can obtain in either country, and getting to that point involves many steps, as briefly described here.
Although Canada and the United States share a border and a language and are cooperative in many ways (including allowing easy border crossings), neither country offers citizens of the other country any special rights to permanent immigration.
Their respective immigration laws do have some commonalities, though. For starters, you will need to fit yourself into some highly complex categories of eligibility in order to become a resident of the other country. Only after you've spent some years as a legal resident and demonstrated good moral character (in particular, committed no crimes) can you apply for citizenship in either the United States or Canada.
If, however, one of your parents was born in Canada but you were born in the United States, or vice versa, you might qualify for citizenship in both countries. Check the laws that were in place during the year of your birth. (See, for example, Automatic U.S. Citizenship for Children by Birth to Citizen Parents (Acquisition).)
If you were not born in the United States and don't have parents who are citizens there, you'd need to take several steps to become a citizen. First, you'd need to apply for a green card or "permanent residence," which will allow you to live in the United States on a permanent basis.
The main categories for green cards are based on either having close family members who are U.S. citizens or permanent residents, having been offered a job in the United States in a field in which the employer is unable to find a ready, able, and willing U.S. worker to fill the position, or having $1 million to invest in a U.S. business. (See this green card quiz for the various routes.)
Once you have been a U.S. permanent resident for five years (at least half of which were spent living in the United States) and met various other criteria, you can apply to become a naturalized U.S. citizen.
The U.S. citizenship application process involves attending an interview and taking exams, during which you will need to prove you can speak and understand English and correctly answer a number of questions on U.S. history and government. You'll also need to take an the oath of allegiance to the United States and be willing to serve in the U.S. Armed Forces if called upon to do so. (Remember, the oath does not mean you can't get dual citizenship, and the United States does not currently have a mandatory draft.)
For more information, see the website of the U.S. Embassy in Canada.
If you were not born in Canada, you will need to apply to become a permanent resident there before proceeding to apply for Canadian citizenship. As with the United States, the primary immigration categories have to do with family, skilled work, and investment/entrepreneurship. Once you have lived in Canada for three of the previous four years, without having been convicted of any crimes, you can apply for Canadian citizenship. (You need not have been a permanent resident of Canada for the whole three years, however.)
Be prepared to take a written test to display your knowledge of either English or French. You will also have to pass an exam that shows you know the history of Canada, as well as the rights and duties of its citizens.
For more information, see the website of the Canadian Embassy in the United States.
As for the "dual citizenship" aspect of this inquiry, there is no separate application procedure by which one must apply for it. If you're already a citizen of either the U.S. or Canada and become a citizen of the other without taking active steps to renounce your original citizenship, you are a dual citizen. It's as simple as that.
Find an attorney who specializes in immigration law within either the United States or Canada to devise an immigration strategy and ensure you are up-to-date on all the country's requirements.
Getting permanent residence in either country is the tough part. Once you've overcome that hurdle, becoming a citizen should present few complications. (After all, you probably already speak English!)