In the process of mitosis that starts from zygote, how do different cells appear? What happens that some cells become one type and some another?
For example, is there a cell that divides into a blood cell and a skin cell?
The process of formation of different types of cells that differ in morphology and function is termed as cell differentiation. In multi-celled organism, each task of life is carried out by different types of specialized cells say nerve cell for transmitting impulses, muscle cells for coordinating movement and these cells differ in their appearance and function, but originated from the same parental cell, the zygote. These specialized cells are formed by a process called differentiation.
Cell differentiation is the process by which genetically identical cells of an embryo become specialized.
Cell specialization involves the preferential synthesis of some specific proteins like antibodies in plasma cells or Hb in erythrocytes.
The genome is the same in all cells of our body, but only certain genes are turned on and other genes are switched off and this lead to specialization. That is, genes active in the neurons may not be active in muscle cells. For instance, genes for actin and myosin filaments are present in all animal cells, but these genes are active only in muscle cells.
I'm not sure about the first developmental stages but, given you already have hundreds of cells with slightly different physiology, the next developmental stages like dev. of neural tube happen through excretion of translation factors and growth factors in several cells. Each of those cells that are in a region where more than one excretion overlaps get a higher dose than the others. This is the signal for the cell to differentiate into another type, by changing its physiology. Such factor excretions are important events, and the genes encoding for the factors are evolutionally highly conserved (so called homeobox proteins).
However, this has nothing to do with mitosis. The newly divided cell does not decide on its own but reacts to a stimulus, like skin cells develop into skin cells because they are on the outside of the cell ball. Excreted factors do the fine tuning.