Once upon a time there was a woman named Karen Keegan who needed a kidney transplant. So they tested her three sons to determine if any of them had sufficiently matching tissue to make a transplant viable.
And the results of the tests came back, and showed that she was not the mother of her children. Now, one might assume some horrible mix-up at a maternity hospital at this point, but it was also clear from the tests that all three brothers had the same father. Further testing of extended family members showed that Keegan's sons were definitely related to her, just not to the extent that she could be their mother.
At first, the doctors were really rather confused. And then they realised that Karen Keegan was a chimera- she somehow had some organs with one set of DNA, and others with another. Her blood sample, which had been used to carry out the genetic test, had the other genes to her ovaries and thus the ones she had passed to her children.
Chimerism is caused by the fusion of fraternal twin zygotes, usually only a few days after fertilisation. The end result is a perfectly "normal" person, and the vast majority of chimeras go undetected. It's now thought that people with two eyes of different colours are likely to be chimeric, and the very rare cases of true hermaphroditism, ie a person with both male and female sex organs, are probably likewise found in chimeric humans.
It is thought that identical twin embryos can fuse as well, but of course this is pretty much undectable if the fusion completes. There is a suggestion that conjoined twins can sometimes be the result of incomplete fusion as well as incomplete splitting of embryos.