The gametes, or oocytes and sperm, are the cells types responsible for reproduction. As such, these cells must retain the ability and information needed to produce all possible cell types in the next generation. They are essentially an unbroken chain of life.
Oocytes and sperm are similar in that they both undergo meiosis to produce haploid cells, but the timing and specific details of oogenesis and spermatogenesis differ between males and females. Additionally, each species often has unique features of gametogenesis.
A summary can be found in the section,The structure of the gametes, from Gilbert's Developmental biology here: https://www.ncbi.nlm.nih.gov/books/NBK10005/
Fertilization might seem like it "just happens" but it is actually a complex process process, some aspects of which are still not entirely characterized.
Review the chapter, The Recognition of Egg and Sperm, from Gilbert's Developmental biology here: https://www.ncbi.nlm.nih.gov/books/NBK10005/
The newly fertilized zygote is one very large cell. The early cycles of cell division are called cleavage, because they cleave the large cell into smaller cells. Different types of embryos have different patterns of cleavage. For example, embryos of different species can exhibit different axes of division, such as meridional or equatorial, or have cleavage that creates cells of equal or unequal sizes.
Holoblastic: the entire cell is divided during cytokinesis; the cells may be equal or unequal in size
Meroblastic: due to presence of yolk, some parts of the embryo are not divided during cytokinesis