How do cells in an embryo become different types of cells?

Different signals can cause embryonic cells to select specific parts of the DNA which can then be used to synthesize proteins, eventually building different cell types.

genetic variation can be caused by mutation (which can create entirely new alleles in a population), random mating, random fertilization, and recombination between homologous chromosomes during meiosis (which reshuffles alleles within an organism's offspring).