Gametogenesis in females happens in ovaries and the formation of gametes in the ovaries is termed as oogenesis.
Oogenesis in females begins before they are even born. It starts at the 6-8th week of the fetus development. At this stage, the primordial (primitive) germ cells starts differentiating into oogonium within the ovaries. They start dividing vigoursly, at around 20th week of fetus development, the amount of oogonium present will be around 6-7 million in number. After 20th week, the oogonium starts degenerating and by the time of birth around 2 millions of oogonium will be found in the fetal ovaries. 
Oogonium, are diploid (2n) stem cells that divide mitotically to produce millions of germ cells. Even before birth most of the cells degenerate in process known as atresia. The oogonium that is survived starts differentiating, where only few of them forms primary oocytes. So at the time of puberty only around 60,0000 to 80,000 primary oocytes are formed and will be in arrested phase of meiosis I prophase (diplotene stage). During this arrested phase each primary oocyte is surrounded by a single layer of flat follicular cells, and the entire structure is called a primordial follicle.
Out of 60,000 to 80,000 primary oocytes at puberty only around 400 will mature and ovulate during the women’s reproductive lifetime. Each month after puberty until menopause, gonadotropins (FSH & LH) are secreted by the anterior pituitary gland which further stimulates the development of several primordial follicles. Out of which only one will typically reach the maturity needed for ovulation and develops into primary follicles.
The primary follicle which was arrested at meiosis I prophase (diplotene stage) starts dividing and forms secondary follicle. The secondary follicle eventually becomes larger turning into mature (graafian) follicle.
Just before ovulation the diploid primary oocyte completes meiosis I producing 2 haploid (ncells of unequal size each with 23 chromosomes. The smaller cell produced by meiosis I is called the first polar body, is essentially a packet of discarded nuclear material. The largest cell known as secondary oocyte receives most of the cytoplasm. After the formation of secondary oocyte it begins meiosis II, but will be arrested at the metaphase II stage.
The mature (graafian) follicle soon ruptures and releases its secondary oocyte, a process known as ovulation. At ovulation the secondary oocytes are swept into the uterine tube, if fertilization doesn’t occur the cell degenerates. If the sperm is present in the uterine tube and one penetrates the secondary oocyte, the arrested meiosis II resumes. The secondary oocyte splits into two haploid cells again of unequal size. The larger cell is the ovum or mature egg. The smaller one is the secondary polar body. The nuclei of the sperm cell and the ovum then unites forming a diploid zygote. 
1. Principles of Anatomy And Physiology 12th edition By Gerard J. Tortora and Bryan Derrickson
2. Langman’s Medical Embryology 12th edition by T.W Sadler