In order to test for Native American ancestry, you will need to have just the right connection between the ancestor you believe was Native American and the person being tested.
Both the yDNA test and the mtDNA test will tell you your haplogroup. The haplogroup will give you an idea of the migrations of your ancestral family from earliest times and can confirm Native American ancestry.
For a female Native American ancestor, you will use the mtDNA test. You will need to locate a person whose mother's mother's ... mother is the targeted Native American. If there is any male in the direct line between the person being tested and this ancestor, you will be testing some other ancestor's mtDNA and will not get what you seek. mtDNA is passed from a mother to her children. Men carry their mother's mtDNA, but cannot pass it on.
For a male Native American ancestor, you will use the yDNA test. You will need to locate a man whose father's father's ... father is the targeted Native American. If there is any female in the direct line between the man being tested and this ancestor, you will be testing some other ancestor's yDNA and will not get what you seek. yDNA is passed from a father to his sons. Women do not carry yDNA at all.
For a step-by-step guide to the whole process, please read DNA the Smart Way http://worldfamilies.net/smart