In the broadest sense, Transgender means "does not identify with the gender they were assigned at birth," but more narrow definitions exist. As such, Transgender is often seen as an Umbrella term under which many Non-Binary and agender people also gather.
In a more narrow sense, it means people who identify with the opposite binary gender than they were ascribed at birth (for example somebody who at birth was called a boy but who is really a woman). Whether or not someone actually transitions or how far they go in that transition is irrelevant. Somebody who was assigned male at birth but who transitions to female is called a transwoman or transfemale/transfemme (as an adjective) and the opposite is called a transman or transmasc (adjective).
To be very very clear:
- A trans woman is a woman
- A trans man is a man
- You are not entitled to know a trans person's old name (deadname) or the state of their transition/operations they may have had.