In a Nutshell
One set of DNA vs Two. Hybrids are animals which occur from the mating of two different species. A chimera is an animal composed of cells with two different DNA sets; they form when two embryos merge and grow as one.
The Whole Bushel
A common, if not wholly accurate, definition of a species is ‘a group of organisms able to breed to produce viable offspring.’ In the case of hybrids they are the offspring of two different species with each of their cells containing DNA from both species. Usually these organisms display features of both species blended evenly together—see Napoleon Dynamite’s favorite animal the Liger, which is the crossbreeding of a lion and a tiger.
A chimera in nature only ever occurs within a species. We are all familiar with twins, two embryos being carried in one womb at the same time. We are even familiar with Conjoined twins, twins whose bodies become attached during development. A chimera results when two embryos merge at a very early stage and the two embryos develop as one being.
This results in one body with two sets of cells and two sets of DNA. In animals this can lead to striking patches of colour as one set produces one type of fur and the other type of cells making a different one.
Chimeras can also be produced artificially in the lab between embryos of two different species. One productive avenue of research being pursued at present is the making of chimeras between human and animal embryos.
While there is no chance that you are a hybrid it is possible that you might very well be a chimera. There have been reported cases of people failing DNA tests because of the presence of a second set of DNA from a twin they absorbed during development. Yep—you could be half you and half your twin.
Show Me The Proof
Colorado State University: Mosaicism and Chimerism
National Geographic: Two Species Become One
National Geographic: Animal-Human Hybrids Spark Controversy
New England Journal of Medicine: Disputed Maternity Leading to Identification of Tetragametic Chimerism