Here are some helpful definitions of some terms you see around this site:

Primary Definitions:

Transgender (trans) - "Identified with a gender other than the biological one." Transgender is a very broad term for anyone that feels their gender identity doesn't conform to the one assigned at birth. Under this umbrella term there are several sub-groups, some described below. There is still some ongoing debate about what does and doesn't fall under this "umbrella" term.

Transsexual - "A person who has undergone surgery and/or hormone treatment in order to acquire the physical characteristics of the opposite sex." Being transsexual isn't about "sexual activity", it's not talking about that (there are many transsexuals who are celibate for example). Instead, it's describing individuals for whom their gender incongruence is so strong, they seek medical treatment through hormone therapy, and sometimes (but not always) physical surgery. The word "transsexual" is more a medical term and is not used very often in casual conversation anymore. "Trans" is more common.

MTF - Male to Female. Describes the direction of transition from male to female.

FTM - Female to Male. (See above)

AMAB/AFAB - Assigned Male At Birth / Assigned Female At Birth

HRT (Hormone Replacement Therapy) - The medical process of artificially modifying the body's hormone levels. In the case of an MTF individual this typically means the lowering of the male hormone testosterone, and supplementation with the female hormone estrogen. The reverse is true for FTMs.

Transition - Transition is the word used to describe the process of moving from one gender to another gender. This process usually takes several years, and can involve the help of supportive family and friends, medical doctors, psychologists, speech therapists, surgeons, and more depending on the individual and their gender "destination".

Full Time - Full time is the time at which a trans person begins living "full time" (24/7) as their target gender.

Cisgendered - "Not transgender; identifying with the same social gender as most people with the same biological sex"  or "Someone who associates and is comfortable with the gender they were assigned at birth."

Note: PFLAG has done a great job putting together an excellent list of helpful definitions. They are a wonderful organization. You can read more on their site.

Other related definitions:

Crossdresser - "A person who wears clothing and other accoutrement commonly associated with the other gender within a particular society." Typically when we talk about crossdressers, we mean a "man wearing female clothing" even though technically females wearing male clothing are crossdressing as well (though this is accepted and even encouraged in our society.) The term crossdresser does not imply anything sexual, though many people assume sexual motives when a man wears female clothes (but not the reverse interestingly). Many men who crossdress will tell you it is not sexual, and 90% of all men who crossdress do not identify as homosexual. They likely do it for the same reason women do - because they like the clothes and nothing more. There is nothing wrong with this, and if you think it's wrong I encourage you to think about the reasons it's demeaning for a man to wear feminine clothes, and not for a woman to wear masculine clothes. There's some implied sexism to explore there. :)
 
Transvestite - "A person, typically a man, who derives pleasure from dressing in clothes appropriate to the opposite sex." This label does imply sexuality. A transvestite is someone who cross dresses as a sexual motivation, or because of a fetish. Most people who wear clothing of the opposite gender are not transvestites. Because of its sexual connotation this word should be considered inappropriate and only used by a person who specifically chooses this label for themselves.