When a zygote is formed by the fusion of sperm (male gamete) nucleus and ovum or egg (female gamete) nucleus it keeps dividing by a phenomenon called mitosis. Mitosis is the division of a parental cell into two new daughter cells keeping the chromosome number constant as any variation could lead to genetic disorders. Initially the cytoplasmic division of the cells results in cells of different sizes. The 16 celled stage is known as morula. The cells grow in size to attain a uniform size.
Later, a mass of cells is formed called the blastula which is a 32 celled stage.The embryo now undergoes gastrulation to form the ectoderm, mesoderm and endoderm which later develop into different parts and organs of the adult animal body.
The cells constantly divide to increase the number of cells. The zygote eventually takes shape of the animal. Also there exist mechanism within the cells that determine cell fate and functions. This is controlled by genes which are expressed spacio-temporally. This nature is very important in animal development. An animal would not be formed, there would not be tissues, no organs if genes were not expressed spacio-temporally. Also some genes control the expression (could either activate or repress) of other genes. There are maternal proteins, obtained from the cytoplasm of the ovum, which are transcription factors involved in transcription of genes.
There operates another mechanism called as apoptosis which is basically programmed cell death. The cells have a life span and various cells are signaled to die. Senescent cells and the unwanted ones die and fresh ones are produced.
So this perfect balance between mitosis and apoptosis ultimately blooms into an organism.