Transsexualism or Transsexuality is a part of transgender. The actual definition of it is debated, some people say it only refers to transgender people within the Gender Binary, others say that it only includes those who actually want to transition medically, and thus includes non-binaries who transition. Others consider it to be anyone whose identity is divergent to their physical body. Thus, a person may be considered transsexual before they make an active decision to be. That would correspond to the definition of "gender dysphoria".
The suffix of "sex" in transsexual (and cissexual) refers to the body, not to the act of sexual intercourse or sexual orientation. Many people mistakenly believe that transsexuality is a sexual orientation, rather than a term related to gender identity.
The most known 'kinds' of transsexual people are transgender women (women assigned-male at birth, abbreviated DMAB) and transgender men (men assigned-female at birth, abbreviated DFAB). However, transsexual neutrois, androgyne, and others are starting to crop up, as well as people who simply refer to their transition as "-to-WTF". At this point in time, the WPATH Standards of Care and DSM-IV do not acknowledge non-binary genders, nor do many gender therapists.
Statistics on transsexual people are very hard to come by and not always accurate. Many transsexual people seek to live in stealth after their transition, as transsexuals face prejudice, employment discrimination, and violence.
A further element of confusion is introduced because "transition" is a poorly defined term. Some people mean the process of preparing and starting to live as the gender-of-identity. This might involve hormone treatments, electrolysis, etc. It does not necessarily lead to surgery, in part because the cost of surgery is prohibitive to many. In particular, "transition" may be statistically defined as genital surgery, and this is a particular issue for transgender men. The very high cost of genital surgery, and ambivalence about results, leads many transgender men to limit themselves to "top surgery" (a double mastectomy) and to forgo genital surgery. This can lead to under-representation in statistics, as well as legal difficulties where laws about gender change are defined in terms of genital surgery.
As a result of this, statistics can be wildly variable and extremely inaccurate. And these are statistics just on people who seek medical transition, people who are comfortable in their body or otherwise don't need to transition are rarely considered in studies on trans people.
One example of how unreliable statistics on transgendered people are, is the statistics on the prevalence of transsexual people who seek genital reassignment surgery (GRS)  . The DSM states "There are no recent epidemiological studies to provide data on prevalence of Gender Identity Disorder. Data from smaller countries in Europe with access to total population statistics and referrals suggest that roughly 1 per 30,000 adult males and 1 per 100,000 adult females seek sex-reassignment surgery" (p. 579). However, the number of genital-reassignment surgeries done to US citizens indicates that roughly 1/2,500 assigned-males have undergone GRS.
Lynn Conway's Transsexual Women's successes http://ai.eecs.umich.edu/people/conway/TSsuccesses/TSsuccesses.html
Lynn Conway's Succesful Trans Men http://ai.eecs.umich.edu/people/conway/TSsuccesses/TransMen.html
Transsexual Road Map http://www.tsroadmap.com/index.html