How do essentialism and/or social constructionism serve as models for gender identity and the formation of gender identity?

Essentialism and constructionism are the two modes of thought that guide our ideas about sex and gender.
It is important to note that sex is a biological fact based on one's chromosomes and the expression of those chromosomes in the development of a fetus. Gender is a social construct—how one chooses to express one's male or female identity, which can also include the expression of a more androgynous identity.
Essentialism tells us that men and women have certain innate characteristics that determine the expression of a particular identity. We are wired to behave in certain ways. While there are scientifically proven differences between the sexes, particularly physical differences, even scientific research is sometimes guided by socially constructed ideas about sex differences. 
For example, the ability to nurture is mainly associated with women. Essentialist ideas about femininity indicate that a woman, due to the presence of estrogen and the release of hormones that encourage bonding with an infant, is more naturally inclined to want to care for her child. Constructionists counter that there are plenty of women who express little to no interest in their children, or those who do not wish to have children at all. Moreover, there are many fathers who wish to nurture and seek roles as stay-at-home parents and early-childhood caregivers but may be discouraged from doing so due to our unwillingness to see these roles as anything but feminine. 
Arguably, much of what we believe about gender identity is socially constructed, though notions about gender expression are expanding to include more models of behavior. Essentialist ideas, traditionally, have focused on gender binaries, which excluded expressions of gender from intersex people. Essentialism, with its belief in fixed essences in men and women, also left little room for acknowledging the existences of transgender people.