Mitochondrial DNA (mtDNA) is genetic material found in mitochondria. It is passed down from mothers to both sons and daughters, but sons cannot pass along their mothers' mtDNA to their children. This is because mtDNA is transmitted through the female egg. The mtDNA found in the egg is nonrecombinant, meaning that it does not combine with any other DNA so that it is passed down virtually unchanged through the direct maternal line over the generations. You inherited your mtDNA exclusively from your mother.
The mtDNA test is more of an anthropological test than a genealogical test. From a genealogical standpoint, mtDNA is not very useful for two reasons. The first reason is mtDNA cannot be tied to any specific surname. Why? Because you inherited your mtDNA from your mother, who inherited it from her mother, who inherited it from her mother, and so on back in time. Traditionally, women in Western cultures change their surnames when they get married, so your mother did not have the same surname as her mother, who did not have the same surname as her mother, etc, etc.
The second reason mtDNA is not very useful for historical genealogical research is because mtDNA mutates very, very slowly - much slower than yDNA. This means that your mtDNA is nearly identical to the mtDNA of your straight line maternal ancestor who lived thousands of years ago, and it is also identical to thousands of people living today. My mother has over 1,000 low resolution mtDNA matches in the FTDNA database. This means she shares a common maternal ancestor with them somewhere back in time. The problem is there is no way to know whether this common maternal ancestor lived recently or thousands of years ago.
Regarding recent kinship, mtDNA works best at disproving relationships rather than proving them. For example, if your mtDNA is not the same as your mother's mtDNA, this means she is not your biological mother and you were adopted. However, it is much more difficult (if not impossible) to tell from your mtDNA alone exactly who your biological mother really is, because your mtDNA is going to match the mtDNA of thousands of women.