Here are some definitions of terms:
Transgender: A transgender person is someone with a gender identity that does not match their sex, sometimes simplistically described as feeling like “a woman trapped in a man’s body” (transgender women) or vice versa (transgender men). The dissonance between that sense of self and body often leads to intense distress, known as gender dysphoria.
The word is an adjective, not a noun. “Transgenders” is thus grammatically incorrect, and the proper terms would be ‘transgender man/woman/people/community’, sometimes shortened to ‘trans’ for convenience.
Transsexual: The subset of transgender people who experience body dysphoria and/or pursue medical transition. However, due to the historical stigma of the word and its connotations with sex, the word ‘transgender’ is more commonly used.
Cisgender: Someone who is not transgender. ‘Cis’ is a Latin prefix that means ‘on the same side of’, while ‘trans’ means ‘across’ or ‘on the other side’. This is more accurate than calling someone a ‘biological’ or ‘genetic’ man or woman. Transgender people who have undergone medical transition are biologically closer to the sex they have transitioned to. Due to intersex conditions, some people may also be born genetically one sex but anatomically another, or mixes of both.
Genderqueer/non-binary: Genderqueer or non-binary people are transgender people who do not identify as either men or women, but as both, genderless, or another gender altogether. Some of them experience body dysphoria and may also seek medical transition.
Genderfluid: People who experience a fluctuating gender identity that can be accompanied by body dysphoria. For instance, someone with a typical female body may be perfectly content with that body and being a woman at some times, but experience periods of time where they are suddenly hit with body dysphoria and a strong sense that they are men and their body is wrong, during which their experiences are very similar to that of transgender men. Genderfluidity has so far been little studied in medical literature, although one 2012 study speculated on possible neurological explanations for what they term ‘alternating gender incongruity’.