Dual citizenship refers to the privileges an individual has as citizens of more than one country. There are only four ways to legally become a citizen of a country. First, if you were born on their soil; second, if at least one of your parents were citizens of that country; third, if you married a citizen of that country; and fourth, if you obtained citizenship of that country through a process of naturalization.
Some countries will have stringent requirements regarding procedure. The United States' process is notoriously difficult to qualify for, as they require American parents to live in the U.S. for a period of time before their children automatically qualify. Furthermore, it's becoming more difficult to gain citizenship in the United States by simply marrying an American.
Other countries have a more lax process. For example, it's perfectly legal to hold a citizenship in both Canada and the United States. It largely depends on the laws of each country. For the most part the country you are applying to doesn't care about the citizenship laws of other countries. When you're in Canada, nobody cares if you're a United States citizen and what legal rights you would have according to that system of justice. The reverse applies for being a citizen of the U.S. while living in the U.S.
Some countries may have restrictions and a very few may require total disassociation from the first country. This is why it's important to verify with all countries just what the naturalization process involves and if there are any rules regarding dual citizenship. While you can expect some tax advantages you also have to prepare for additional responsibilities. You will have to pay taxes for all the income you make from each respective country. You will also have to follow the law regarding military service and any travel restrictions listed.
The advantages of dual citizenship include more flexibility as far as employment and homing. You also have more financial stability than when you depend on just one country entirely for your income and investments. If you are seriously considering applying for another citizenship then it's a shrewd idea to speak with an attorney that is familiar with immigration laws. You can also speak to consular officials. If you don't wish to communicate these offices then you should at least verify your plans with an authoritative source. Immigration law can be complicated and costly, especially if you are unprepared.