A trans woman (sometimes trans-woman or transwoman) is a woman who was assigned male at birth. The label of transgender woman is not always interchangeable with that of transsexual woman, although the two labels are often used in this manner. Transgender is an umbrella term that includes different types of gender variant people (including transsexual people).
Sex assignment refers to the male or female category that a newborn baby is placed in, usually based upon the appearance of external genitalia.
Gender identity refers to a person's private sense of, and subjective experience of, their own gender. This may be different from the sex and gender that the person was assigned at birth.
"Transition" refers to the process of adopting a social and personal identity that corresponds to one's own sense of the gendered self, and may or may not include medical intervention (hormone replacement therapy, surgery, etc.), changes in legal documents (name and/or sex indicated on identification, birth certificate, etc.), and personal expression (clothing, accessories, voice, body language).
Both transsexual and transgender women may experience gender dysphoria, distress brought upon by the discrepancy between their gender identity and the sex that was assigned to them at birth (and the associated gender role and/or primary and secondary sex characteristics).
Both transsexual and transgender women may transition, though only transsexual women would medically transition. A major component of medical transition for trans women is estrogen hormone replacement therapy, which causes the development of female Secondary sex characteristics (breasts, redistribution of body fat, lower waist–hip ratio, etc.). This, along with sex reassignment surgery can bring immense relief, and in most cases, rids the person of gender dysphoria.