As I understand it, trans means "an individual whose gender identity is different than what they were designated at birth". However, I also hear the terms transgender and transsexual used for similar meanings. I know that personally I only refer to myself as trans or rarely transgender, and never transsexual, and the latter sounds very odd to me. I'm interested in why there are several words for roughly the same meaning, and if they do have different meanings or simply based on geography and time period.
Transexualism refers most specifically to gender identity, whereas transgenderism encompasses gender expression as well (cross-dressing for instance); that is, transexualism is a particular form of transgenderism.
For a number of different reasons that are probably well outside the scope of an answer on SE (seriously, I'm sure there are people who have written entire books on the subject), there are a proliferation of terms that one or another people prefer, feeling that previous terms don't best fit their particular circumstance — compare with feminism, womanism, womynism, etc.
trans is often used as a shorthand for those terms that begin with trans. Due to their number, rather than create an unwieldy acronym like LGBTQQIAAP, you'll often see trans* used to refer to the entire community, where the asterisk acts as a wildcard effectively saying "insert each person's prefered term here" to be at the same time inclusive and brief.