A lot of people whom Zoeb refers to as transsexual above, are now also referred to as transgendered.
That is, in general, 'transsexual' may be used to describe a person who has transitioned, whereas 'transgendered' refers to all persons who have have gender issues, Gender Identity Disorder and/or are other gendered, as well as transsexuals as defined above.
However, a movement has arisen for some post-transition people to call themselves 'persons of transsexual history' or 'persons who have experienced transsexualism'. The idea is that 'transsexualism' is a condition you have, not the person you are, and should be treated linguistically and clinically in the same way as schizophrenia or diabetes or epilepsy: You are a 'person who has schizophrenia', you're not 'a schizophrenic', etc. Likewise this medical model of transsexualism recognises it as a medically treatable condition, and some groups will even go so far as to say that a person who has transsexualism will require medical treatment (Sex Affirmation Treatment) in the form of hormone therapy and or surgery (aka Sex Reassignment Surgery/SRS or Gender Reassignment Surgery/GRS) in order to survive.
Within this model comes the concept of 'brain sex', which is the concept that our brains have a sex, independent of our bodies. Usually, brain sex and body sex match up, but sometimes, due to reasons yet unknown (perhaps a mis-timing of the hormonal wash in early foetal development?), the brain develops a sex different to the body, therefore producing the condition of transsexualism in a person, which can be seen as a type of intersex.
In this medical model, a person who has transsexualism is a person who will, or has transitioned - and once has done so ceases to be transsexual in the present (thus becoming a person of transsexual history...). A person, therefore, who does not necessarily want or need to physically transition, and/or who sees hirself as a gender other than male/man or female/woman; or a person who transitions, lives wholly within 'man' or 'woman' and still acknowledges him or herself as trans*, is by definition not transsexual.
Marriage:
In Australia, it is legally possible for a transsexual person to get married. This relies on the premise that a post-transition transsexual person is legally the sex s/he presents him or herself, and therefore can marry a person of the opposite sex. The precedent for this was set in a case known as 'Re Kevin', which was fought and won on 12 October 2001 in the Family Court of Australia, a decision which was upheld on appeal in the same court on 23 February 2003.
Ref: http://www.austlii.edu.au/au/cases/cth/family_ct/2001/1074.html