If you take two individuals, and trace any particular piece of their genome—say, some small stretch of chromosome 21—back in time, eventually you will end up finding that both individuals have inherited that particular bit of their genome from a single individual. If they are identical twins, that will happen only one generation ago; if they are cousins, it could happen as soon as two generations. But no matter how unrelated the individuals are, eventually it will happen in all parts of the genome, including the mitochondrial DNA and Y-chromosome.
The same applies to the whole population. If, for example, you sequenced everyone's mitochondrial DNA, and built a pedigree representing the ancestry of the people, you would end with a single great-great-great-great-great-etc. grandparent — the so-called most recent common ancestor (MRCA). If you repeated the process with a different bit of the genome, the Y-chromosome for instance, you might get a different time to the MRCA.
How long the process takes — in other words, how far back to the MRCA — depends on the history of the population and how big it is. (The exact details of the estimates also depend on what you assume about generation time and how fast genetic changes accumulate, but that's the basic idea.) For technical reasons, because of the way the process works, you don't actually have to sample everyone in the population, but can be reasonably confident about estimates of the time to the MRCA with small samples, as long as your assumptions are right.
So who were mitochondrial Eve and Y-chromosomal Adam? Basically, they’re cutesy names for the MRCAs in those particular parts of the genome. The reason they have names (and there’s no chromosomal Charlie, or what have you) is that the mitochondria and Y-chromosome are particularly informative and unique bits of the genome. Because they are passed on through only one sex and are inherited as intact units, they carry more information than the rest of the genome, which gets switched around and reshuffled with each generation.
All that is to say that you are descended from mitochondrial Eve through your mother, grandmother, great-grandmother, etc., and so am I! That doesn't mean that she was the only woman alive at the time, just that she was the only one alive at the time who has left an unbroken line of female descendants until today. Obviously, her line of descendants continues back through time, through her mother and grandmother, you get the drift, which is why she is the most recent common ancestor, and not just the common ancestor.
For the genome as a whole, however, it doesn't make sense to talk about a single common ancestor. Because the genome gets shuffled and broken up between generations by recombination, you inherit different bits from different ancestors. As a result, one piece of the genome will likely have a very different history than some other. And although each particular piece of the genome can be traced back to a common ancestor, no one historical person contained the entire intact ancestral genome. That is to say, the current population has lots of MRCAs. If we were asexual clones, however, there would indeed be a historical person that was the common ancestor for the whole genome.
The identical ancestors point (IAP) is something I know much less about, but I did a bit of reading (see Nature 431, 518–519; 2004). On a technical level, it seems that the IAP, unlike the MRCA, has to do with relatedness by pedigrees, and not by genetics. To illustrate the difference, consider the probability that you are not genetically related to your grandfather for some particular chunk of the genome: you only get half of your parent's genes, and they only got half of their parents. So, if you do the math, you have a 75% chance of not inheriting any particular bit of your genome from your grandfather, but chances are you inherited some amount of DNA from him. The probability that you didn't get any genetic material from a particular ancestor increases for more and more distant ancestors; however, in a pedigree sense, these people are still your ancestors, just as your granddad is still your granddad.
Now, setting the genetics aside, if you trace pedigrees back through time, eventually you come to a common ancestor — someone to whom we are all related. Because we can have more than one ancestor, there can be more than one of these people. The further back in time you go, the more ancestors we all have (2 parents, 4 grandparents, 8 great-grandparents), and the fewer people there are to be these ancestors. Thus, you end up with more and more of the common ancestors in the population.
At some point, there will only be these common ancestors, and people who left no descendants today — this is the IAP. This time point is further back than the MRCA, because the MRCA's ancestors are also our ancestors — mitochondrial Eve's mother was also a common ancestor, after all, just not the most recent common ancestor.
[Editor: There's a good article on the subject with diagrams, at scienceblogs.com]
making a small donation to science.ca.