TGI is an "umbrella term" intended to cover a range of identities including the broad identity terms of transgender, transsexual, gender-variant, and intersex. Within these broad terms are the identities male-to-female (MtF), female-to-male (FtM), transwoman, transman, genderqueer, androgyne, bigender, genderf*ck, Two-Spirit, and many others. In general, people who identify as TGI feel that the gender they were assigned at birth is not an accurate or complete description of their gender identity.
A doctor declares a baby's sex (male or female) by looking at the child's external reproductive anatomy when the child is born. Sex can also be determined by examining internal reproductive anatomy, hormone levels, or chromosomes. Often, all of these things match. Sometimes they don't, or one or more of them is not clearly male or female; when this happens, a person may be considered intersex. Sometimes parents and doctors choose to perform surgeries on babies who are intersex to "make" them anatomically male or female.
Gender identity is a person's inner sense of being a man (or boy) or woman (or girl). Many males have an inner sense of being a man, and many females have an inner sense of being a woman. Sometimes, though, the physical sex and the inner sense of gender identity don't "match," so that a male person feels more like a woman inside or a female person feels more like a man inside.
Gender expression refers to the way people present their gender to other people in society. A common indicator of gender is the clothing a person wears, but hairstyles, body characteristics (muscle tone, broadness of hips or shoulders, skin texture, etc.), and even speech patterns and posture are also part of a person's gender expression. Gender expression can be described as masculine, feminine, androgynous. It is not permanent -- a person who wears a suit and tie to work and then wears a dress and makeup at home in the evening is switching from a masculine gender expression to a feminine gender expression.
All of these things are different from a person's sexual orientation, which refer to whether a person is sexually attracted to men, women, both, or neither.
Western society tends to assume that all males identify as men and will have a masculine gender expression, and that all females identify as women and will have a feminine gender expression. These assumptions are imposed on people starting in the very first moments of life: people assume that a female child will feel like a girl and that a male child will feel like a boy. Society essentially assigns a "sex-appropriate" gender identity and expression at birth based on whether a child is male or female, because most male children do feel like boys and most female children do feel like girls. For many people, this feels natural and comfortable, and they never have to think about whether their gender identity matches their sex, because it does.
TGI people, on the other hand, often do not fit so easily into this set of assumptions. A TGI person may have been declared (or assigned, if intersex) female at birth, but not feel like a girl or woman. The person may feel like a boy or a man, or like some combination of man and woman. The feeling can be hard to describe to people who have never felt it, but Jennifer Finney Boylan, in her book She's Not There, describes it this way:
"It seemed obvious to me that this was something you understood intuitively, not on the basis of what was between your legs, but because of what you felt in
your heart. Remember when you woke up this morning -- I'd say to my female friends -- and you knew you were female? That's how I felt. That's how I knew."
Some TGI people feel extremely uncomfortable living as a person of the gender they were assigned at birth, and decide to transition from one mode of gender presentation to another (for example, from living as a man to living as a woman). This transition may include coming out to other people, legally changing one's name and sex on identification, changing one's body through hormones or surgery, or some combination of these.