A chimera is a genetic organism which has cells from more than one distinct egg. This can be triggered in a few different ways, but it tends to happen due to some mutation in the cell division early on in forming animal. One way which this can happen is if one twin absorbs another very early on in their development which can produce an animal which will have more than one set of DNA. Human chimeras tend to not be aware that they are chimeras unless they take a DNA test at some point in their lives.
While cases are rare, shows like House or CSI have done episodes based on the condition because of the strange legal situations it can throw up. In 2002 an American named Lydia Fairchild was in the process of separating from her boyfriend Jamie Townsend. As part of her application for child support for her three offspring (one of which was still a fetus), Fairchild had to submit DNA evidence that Townsend was the father of the children. The results came back that he was the father, but she was not the mother.
Fairchild was taken to court on a count of benefits fraud, but she was insistent that she was the mother. The judge ruled that there should be a witness present for the birth of the third child and they did another DNA test as soon as it was born. Once again the test showed that she wasn't the mother. Clearly something weird was going on.
It turned out that Fairchild was a chimera, carrying two sets of DNA. Skin samples and cuttings of hair had provided the DNA used for the tests did not match the DNA of the children, but later results from a cervical smear test did match. In a sense she was her own twin. Further tests on the extended family showed that Fairchild's extended family all had the regular proportions of DNA match with the children, it was only Lydia Fairchild herself that threw out the ratios. For instance the children's grandmother had 1/4 of the DNA matching on average.