Mosaicism denotes the presence of two or more populations of cells with different genotypes in one individual who has developed from a single fertilized egg whereas chimerism denotes the presence of two or more genotypes arises from the fusion of more than one fertilized zygote in the early stages of embryonic development.
Mosaicism refers to heterogenous expression of a disease at the cellular or tissue level, resulting from cell-specific differences in the expression of a mutation or the presence of a chromosome aberration. This occurs when two or more distinct genotypes or chromosome complements exist in different cell populations. Mosaicism can result from an error in mitosis after fertilization and the initiation of development (termed postzygotic mosaicism). As a consequence, the affected individual is mosaic, with some cells carrying the mutation, while others do not.
Germline or gonadal mosaicism refers to the finding of normal and abnormal cells with a mutation or chromosome aberration in the eggs or sperm prior to fertilization. Germline mosaicism is important in some autosomal dominant disorders, in which recurrence risks for a further affected child should take the possibility of germline mosaicism in a parent into account, even if both parents appear to be healthy.
As an example, a mutation occurring at the eight-cell stage would (theoretically) lead to one of eight cells in the body carrying the mutation. The expression of disease phenotype will then depend on which cell types harbor the pathogenic mutation. X-linked diseases can manifest mosaicism due to cell-specific inactivation of one X chromosome.
Tetragametic chimerism is a form of congenital chimerism. This condition occurs through the fertilization of two separate ova by two sperm, followed by the fusion of the two at the blastocyst or zygote stages. This results in the development of an organism with intermingled cell lines. Put another way, the chimera is formed from the merging of two nonidentical twins (although a similar merging presumably occurs with identical twins, but as their DNA is almost identical, the presence would not be immediately detectable in a very early (zygote or blastocyst) phase). As such, they can be male, female, or hermaphroditic.
I would try to make it simple by using examples. Consider a patient having leukemia undergoes a bone marrow transplant. He has all cells of his own genotype, but the bone marrow cell progeny will have genotype of the donor. So, at a point his body will have cells having two different genotypes most of cells will have his genotype but, blood cells will have genotype of donor. He is hence a Chimera i.e. an individual having cells of two or more different genotypic origin.
But, if a person had suffered a mutation in one of blastomere (cell) when he was an embryo at eight celled stage, the progeny of seven cells will be normal, but those of eighth cell will be little altered in terms of base composition or may be other aspects. Never the less, all cells including mutated ones are genetically of same individual. This kind of structure having both normal and mutated ones in same body is called mosaicism.