Microchimerism, as defined by wiki, is the presence of a small number of cells in an organism that are genetically distinct, and originated from a separate organism. The example they give is an unborn child and the mother can 'swap' immune cells, and retain these for many years.
Your question about which parts of the genome are transferred is therefore not really valid, because the cells are transferred, rather than bits of genetic information. Therefore 'traits' are not transferred either, but distinct immune cell lineages can be transferred and persist - this can confer an immune advantage, so this 'trait' could be said to have transferred, but no genetic information has been swapped by the cells.
Although it does not say it on wiki, this presumably only applies to cells of the same species, but a different organism, otherwise infections would also count.
Update 20th July 2012
The New Scientist article you have linked to in the comments can be found online here.
It says that we may be more microchimeric than we imagined, because a recent study in Blood found that half of the mothers studied were positive for male T-cells (Dierselhuis, 2012). The authors note that it is remarkable that the immune cells from neither the parent or the offspring attack the other, and that this may explain the observation that siblings make better donors.
Another finding that mothers are found positive for cells from their own mothers, and even their own grandmothers, and that the cell numbers increase during pregnancy (due to clonal expansion of the inherited immune cells) reinforces the opinion that microchimerism is commonplace (Gammill, 2011).
So to reiterate, immune cells can be inherited from your mother, and she may also end up with leukocytes from her offspring, which can in turn be passed on. This is a very fascinating way in which the immune system seems even more complex and helpful.