When I'm explaining my research to laypeople, I usually start out with something along the lines of "With a few exceptions, every cell in your body has basically the same genome." The obvious exceptions are usually places where genome modification plays an integral role in the differentiation process itself -- for example, B and T cells in mammals, whose rearranged receptor loci are responsible for antibody diversity, or the massive numbers of copied, lined-up (polytene) chromosomes in Drosophila salivary glands. In each case, the rearrangement is specific: you don't see rearranged immunoglobulin genes anywhere but in B cells, and you don't see polytene chromosomes outside of salivary glands. These are the mechanisms that the body uses, in certain very rare occasions, to tell one cell from another in a permanent way, or to leverage a genome into doing something that it might not otherwise do. Most cells, on the other hand, are differentiated by chromatin state, protein expression, and other mechanisms which are relatively stable, and relatively permanent, but don't touch the genome sequence below. That is the basic reality which allows a skin cell to be reprogrammed by a certain set of transcription factors into a stem cell: the modifications may be different - the clothes may be different - but underneath it all it's still the same genome - it's still you.
Of course, there's another kind of genome instability that's covered in every genetics class: this is the instability caused by transposons, or jumping genes. This kind of instability is the bad kind, the kind that is unregulated, random, can alternately do absolutely nothing or fundamentally alter the expression of important genes. They are semi-autonomous genetic parasites; the remnants of long forgotten viruses which live on in our DNA; the rats in the lower deck whose numbers must be kept down for the health and safety of the ship itself. Most organisms go to extraordinary lengths to keep these transposons silent. DNA methylation seems to have been evolved for this purpose: by chance, organisms which put a chemical modification onto repetitive sequences of DNA, thus shutting it down, were less likely to be prey to transposon insertions and thus had more stable genomes and better reproductive fitness. DNA methylation of transposon sequences is the one constant of DNA methylation; even in Drosophila, transposon sequences are highly methylated, while the rest of the genome shows no sign of the modification whatsoever. Transposons are reactivated very rarely: a few seem to be important in placental development to do typically viral things like repressing the mother's immune response, and in one example in plants, transposons are reactivated in the endosperm in order to provide RNA templates for RNA-mediated DNA methylation in the embryo (I think). Notice one similarity for both of those examples: transposon reactivation occurs in extraembryonic tissue, not embryonic tissue. What a thrill, then, to learn that this 'bad' kind of genome modification could serve an entirely different purpose.
A paper published in Nature this week shows that L1 elements (a certain kind of transposon that happens to be particularly prevalent in humans), and in particular human L1 elements, are unmethylated, expressed, and capable of insertion in human neural progenitor cells (NPCs). They go on to show that brain tissue contains more copies of the L1 sequence than heart or liver tissue. What does this mean? It means that your brain is likely a genetic mosaic, with different cells harboring these L1 insertions in different numbers and different places. Some insertions could do nothing, and some insertions could kill the cell (in which case they would be lost), but some insertions could prove disabling-but-not-deadly, and some insertions could be advantageous. And since different neurons might have different insertions in different places, this means that there might be a(n epi)genetic reason why different people think in different ways. Even better, this process would be ongoing in neural progenitor cells -- as your brain develops, it is changing and redefining itself by which cells have insertions in which genomic loci. And, since the entire thing happens in the brain, none of these rearrangements are passed down to any children you may have, which might go some distance to explain the non-heritable portions of intelligence. Finally, and perhaps beautifully, as NPCs accumulate these insertions over time, they will, by chance, pick up an insertion in some essential gene that causes their own death (there's a Russian roulette metaphor there that, for once, I will spare you).
The fact that the neural connections in your brain are a plastic, ever-changing structure has been known for quite a while. The idea that the very genome of your brain could also be plastic is, as far as I know, fairly new.
Aside: Personally, I think that this is one of many similarities between extraembryonic tissue and neural tissue, which is yet another reason why the placenta is a very interesting thing to be studying. Also, if you want a cool genetic phenomenon, you should look up "chorion gene amplification" in Drosophila and perhaps "endoreduplication" just in general (it happens in mammals too -- in the placenta, as a matter of fact!).
Edited to add: If you want to read the editorial in Nature, go here. Or, if you want to read the original paper, it can be found here.