"Mitochondrial Eve" is the name given by scientists to the woman who is the matrilineal most recent common ancestor of all living humans. That means that all living humans have mitochondrial DNA derived from hers. Mitochondrial Eve is thought to be a woman that lived in eastern Africa about 140,000 years ago. The concept is distinct from most recent common ancestor -- because the number of ancestors one has goes up exponentially as one travels back into time, the most recent shared common ancestor of all humanity lived more recently, probably less than 100,000 years ago. But due to sexual recombination of nuclear DNA, every living person today has only a fraction of shared genetic material with that common ancestor.
The tests that let scientists know approximately when Mitochondrial Eve lived are based on mitochondrial DNA, small loops of DNA found in the mitrochondria, the power plants of our cells. Mitochondrial DNA only engages in genetic recombination with itself, meaning that it stays relatively constant from mother to offspring. The only factor that makes it change is mutation, and since the average mutation rate is known, scientists can estimate how much different human (or animal) lineages are related to one another, and when their most recent common ancestor is likely to have lived. The estimates tend to be more accurate when the timescale is relatively short, in hundreds of thousands of years rather than millions of years. Timescales involving hundreds of millions of years are generally considered to have low accuracy.
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There are a few misconceptions associated with the concept of Mitochondrial Eve. This woman was not the only human living at the time -- since humans descended from other hominids incrementally, scientists have opined that there was no "Adam and Eve" -- no founding couple, and Mitochondrial Eve only represents the person that all living humans got their mitochondrial DNA from. Mitochondrial Eve was one female among tens or hundreds of thousands that lived in Africa at the time. Other females living at the time may have been the common ancestor of all humans alive today, but there exists at least one male in their ancestral line between then and today. Conversely, Mitochondrial Eve gave birth to a daughter that gave birth to another daughter, through thousands of generations until today, where the mitochondrial DNA is still passed on in a daughter-to-daughter format.