A person’s sex is defined by the appearance or anatomy of their genitals, or their chromosomal make-up at birth. Sex is usually recorded as male, female, or intersex (although intersex itself comprises a variety of different intersex conditions). If you need to talk about the birth sex of a transgender or intersex person, it is best to say 'assigned female at birth' or 'assigned male at birth'. Avoid phrases like 'born male', 'born a woman', etc.
A person’s gender relates to their internal sense of identity. Gender identity can be male or female, agender (no gender at all), non-binary (somewhere between the binary male and female genders), or something else entirely.
A transgender person is someone whose gender identity is different to the sex they were assigned at birth. A cisgender person is someone whose gender identity is the same as the one they were assigned at birth. Gender identity is a spectrum; a natural variation of human identity. It is innate and cannot be forcibly changed, although someone’s journey and expression of their gender identity may change naturally over the course of their life time.
People usually use pronouns (the words we use to refer to ourselves) that correspond to their gender identity. Trans women (or transfeminine people) usually use 'she/her' and trans men (or transmasculine people) usually use 'he/him', while others may use 'they/them' or alternative pronouns. If in doubt, it never hurts to ask how someone would like to be addressed.
Although gender identity is often expressed in the way we present ourselves (how we dress, how we refer to ourselves, how we behave etc.) they are not the same thing. For instance, a trans woman who does not like to wear dresses is still a trans woman.