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Transgender is an umbrella adjective describing people whose gender is other than the one they were declared to be at the time of their birth. Under the umbrella are trans men, trans women, bigender people, those of the third gender, those who don't identify with any gender, and many more.
To be absolutely precise, transgender is an adjective describing people who self-identity with a gender other than their medically designated gender (which often but does not always fall into the gender binary). Transsexual is an adjective describing people who have transitioned to living with a different gender than that assumed at birth, or desire to do so. Some people, however, find the word 'transsexual' offensive, due to its associations with the historical maltreatment of transgender people by the medical profession. 'Transsexual' is sometimes even treated as a reclaimed slur, so if one is cisgender, it is safer to err on the side of caution and use 'transgender' when describing trans people.
Transgender people can be of any sexual orientation, although it is common among transgender people, especially those who are non-binary, to use terms like "androphile" and "gynephile" rather than "heterosexual" or "homosexual," because the latter terms require that one partner be of one gender identity and the other partner be of the opposite/identical gender identity, which may not be possible if one partner self-identifies outside the gender binary.
Transgender is an adjective; thus, the correct form is "transgender people." When talking about transgender men, women, or people, the correct short form is 'trans man', 'trans woman', or 'trans person'; not 'transwoman', 'transman', or 'transperson'. After all, you wouldn't say you wore a 'bluehat' or 'redshoes' - unless, of course, you were German.
Go forth and be inadvertently offensive no more.