In order to define transgender you have to have a working definition of these terms: sex, gender, gender identity, gender dysphoria.
IT'S A BOY, IT'S A GIRL
Gender-reveal parties reveal the view of society that has been accepted up until recently. Gender has been connected to the sex of a baby discovered in an ultrasound. While gender is not a new word, it has taken on additional meaning and usage.
SEX, GENDER, AND GENDER IDENTITY
Sex can refer to our biological makeup. Men have XY chromosomes while women have XX chromosomes. Hormonal, anatomical and reproductive differences stem from chromosomal differences.
Primary sex characteristics refer to differences in reproductive systems. Secondary sex characteristics refer to general differences between men and women. Men tend to have (not always) broader shoulders than women. Women generally have wider hips and might be shorter.
According to the American Psychological Association, gender is "attitudes, feelings, and behaviors that a given culture associates with a person's biological sex. Behavior that is compatible with cultural expectations is referred to as gender-normative; behaviors that are viewed as incompatible with these expectations constitute gender non-conformity."
In traditional society, which includes practically all societies until the last 10 years, if your sex was female that so was your gender.
Each culture has its own expressions for gender. There is no one right or wrong way to express gender. Kilts might be an example where it's masculine to wear one in one culture, but in other culture it might be mistaken for a dress and therefore deemed feminine. There's nothing inherently masculine or feminine about clothing made that way.
What has changed today is that many people see gender as something disconnected from sex. In this view your sex might be female but your gender could be male. Gender is now viewed as a psychological reality as opposed to a biological reality. Scripture doesn't posit such a view.
Because of this changing view, many people that identify as a different gender than their birth sex are opting for surgery to change their bodies to appear as a different sex to match their perceived identity. Gender identity is one's self-perception of whether they are male or female.
All of us have a gender identity. When someone perceives that their gender does not match their biological sex and this causes inner turmoil, that person is said to have gender dysphoria. This is a genuine experience. Some people really do feel as thought their body is lying while their mind is telling the truth. But feelings don't always comport with reality. That feelings is just a feeling.
GENDER DYSPHORIA AND TRANSGENDER
People with gender dysphoria should not be labeled freaks or perverts. It is not something we can just tell people to "get over". Experiences with GD may also differ from person to person. There are degrees of dysphoria from mild to severe. People who identify as transgender report higher rates of mental-health problems as compared to the rest of the population. Problems such as depression and suicide are more common. Compassion needs to be shown to them. Not all those with gender dysphoria identify as transgender. Some with GD identify with their biological sex. GD is an experience which is different than how one might identify. It is when one follows that desire than one is said to be transgender. Transgender is a term that is a catch-all for anyone who deems that their gender does not match their biological sex. Some decide to dress in ways that are considered normative for a particular sex. Others may decide to undergo hormonal treatment to try and bring their chemical balance in line with the sex they believe themselves to be. Some may undergo surgery to try and take on the appearance of a different sex. For some, it may mean being fluid in their gender--not identifying as one gender at all times. Some choose to be labeled as non-binary--not identifying as male or female.
TWO QUICK QUESTIONS
Does anything cause gender dysphoria? There is no definitive answer from a medical perspective. Biblically we would say that this is a result of the fall. Nevertheless, aside from that generality, there's no definitive research that explains why some people have GD.
How common is transgenderism? One institute believes that 0.6% of the US populations is transgender (about 1.4 million people). This will likely increase in a society that is socially tolerant of this view.