Transgender is an umbrella term used to describe people whose gender identity or gender expression do not match the gender they were assigned at birth. For example, some people who were assigned to be male at birth are female (trans women). Some people who were assigned to be female at birth are male (trans men). Some transgender people have medically transitioned, undergoing gender affirming surgeries and hormonal treatments, while other transgender people do not choose any form of medical transition. There is no uniform set of procedures that are sought by transgender people that pursue medical transition. Transgender people may identify as female, male, or nonbinary, may or may not have been born with intersex traits, may or may not use gender-neutral pronouns, and may or may not use more specific terms to describe their genders, such as agender, genderqueer, gender fluid, Two Spirit, bigender, pangender, gender nonconforming, or gender variant.
Nonbinary people have gender identities and/or gender expressions which fall outside of the dominant societal norm for their assigned sex, is beyond genders, or is some combination thereof. Some people use the term Gender Queer to describe this identity. Queer is a term that is offensive to some when used as a derogatory term. Others have reclaimed and self-defined the word as a form of empowerment.
An intersex person is someone whose sex a doctor has a difficult time categorizing as either male or female. It could also refer to a person whose combination of chromosomes, gonads, hormones, internal sex organs and/or genitals differs from one of the two expected patterns (i.e. male or female). Another way of thinking about it is Intersex refers to a series of medical conditions in which a child's genetic sex (chromosomes) and phenotypic sex (genital appearance) do not match, or are somehow different from the "standard" definition of male or female.