"Transgender" is typically used as an umbrella term to describe someone who doesn't adhere to the stereotypical attributes society ascribes to gender. Unlike sex that's biologically and chromosomally determined, gender is a social construct that places men and women into distinct categories, each with its own attributes such as clothing, hair style, mannerisms, speech, walking, etc. Some people don't fit into this dichotomy and prefer to blend gender or, in the case of transsexuals, transform their bodies to match their internal gender identity. Once considered malleable, gender identity is now accepted to be hardwired before birth.
Crossdressers (or transvestites) are usually heterosexual males who express their feminine side, and sometimes get sexual satisfaction, by dressing in women's clothes. Crossdressing is not a disorder and cannot be "cured." Crossdressers don't take female hormones, except for experimentation, nor want sex change surgery.
Drag queens are usually gay men who impersonate women and sometimes perform in night clubs. They typically employ exaggerated mannerisms and garish makeup.
Gender queer, or androgynous, are people who exhibit attributes, in varying degrees, of both genders, or choose not to follow society's stereotypical gender expectations.
Intersexed people, or hermaphrodites (an older term), are born with ambiguous genitals, or a combination of male and female genitalia, internal and / or external. In the past, an intersexed child would sometimes be "surgically assigned" (usually to female), without their consent, in the hopes that they would grow-up comfortable in their assigned gender. These assignment surgeries have been largely unsuccessful as the individual would usually develop the gender identity opposite to the one in which they were raised.
Transsexuality goes much deeper. A transsexual has a deep, core identity with the gender opposite of their born sex. The traits can express themselves very early in life, as young as 3 or 4. Though the cause is unknown, it's thought to happen prenatally and caused by a mix-up in fetal hormone balance at critical times in the fetus's development and could have genetic causes.
A fetus' brain is female by default. If testosterone is produced the brain develops male; if estrogen is produced it remains female. Just as in homosexuality, transsexuality is not a choice. Attempts to change the mind to match the body have failed. The only effective treatments for transsexuals is changing gender through hormone therapy and sex reassignment surgery.
Gender identity and sexual orientation, sometimes confused with each other, are separate and distinct. A transsexual woman isn't necessarily attracted to men and vice-versa.