As the article states, it is..."animals that have more than one genetically-distinct population of cells". Most people/animals/etc have one set of genes. These tell the body how tall to grow, what color eyes/hair/skin to have, how many legs they need, and any other number of physiological and even psychological things. In Mosaics and Chimeras, there are more than one set of these genes flying around their system. In some cases, it even becomes outwardly apparent with changes in skin pigmentation, hair, etc.

http://www.vivo.colostate.edu/hbooks/genetics/medgen/chromo/mosaics.html
"Mosaics and chimeras are animals that have more than one genetically-distinct population of cells. The distinction between these two forms is quite clearly defined, although at times ignored or misused. In mosaics, the genetically different cell types all arise from a single zygote, whereas chimeras originate from more than one zygote. "

This actually happens in humans from time to time:
http://www.cronaca.com/archives/001331.html
"Eight years ago in Britain, a boy was born who, genetically, was two people. He was formed when two eggs, fertilized by two different sperm, fused into one embryo inside his mother's womb.

He was an unremarkable baby. But as a toddler, doctors discovered that he was a hermaphrodite - what was originally diagnosed as an undescended testis turned out to be an ovary, a fallopian tube and part of a uterus. Further investigation revealed that some parts of his body were genetically female but the rest, which contained a different combination of his parents' genes, was male."

http://www.thetartan.org/2008/9/15/scitech/healthtalk
"The fundamental rule of genetics states that parents pass genes onto their children, but recently, a 52-year-old woman from Boston was told that she was not the biological mother of her children, after years of raising them. The case, reported in The New Scientist, has the medical community at its wits’ end searching for an answer.

The woman who claimed to have given birth to her children had a set of genes completely different from those of two of her three children. What was even more intriguing about this case was the fact that the DNA of the children matched their father’s DNA.

After working for nearly two years on the case, doctors came up with an answer to the problem: the woman was a “human chimera.”

I'll try to find a picture of a baby boy who had checkerboard pigmentations on his body as a result...

In genetic medicine, a mosaic or mosaicism denotes the presence of two populations of cells with different genotypes in one individual, who has developed from a single fertilized egg. [1] Mosaicism may result from a mutation during development which is propagated to only a subset of the adult cells.