A transsexual is someone who feels her/his true gender is not the same as her physical sex.
The term transsexual was coined by Dr. Harry Benjamin in the 1950s and used in his book The Transsexual Phenomenon.
The DSM-IV prefers the term Gender Identity Disorder or GID, insinuating that there is something "wrong" with anyone who feels her body's sex doesn't match her mind's gender. It seems to have a bit of trouble differentiating between gender and sexuality too. In fact, many surgeons who mutilate intersexed babies believe they were successful only if the child grows up into a healthy, straight adult of a binary sex and gender.
Transsexuals used to be given electric shock "therapy" to try and convince them they were really deluded about their gender. Needless to say, this method yielded no positive results.
There seems to be evidence now showing that transwomen (Those that identify as women despite being biologically male) have the same sized hypothalamus as genetic females, which is drastically different to that of males. This means there may be some truth to the notion that transsexuals really are the gender they believe themselves to be.
The basic life of a transsexual is something like this: You're born, and someone says "It's a boy!" or "It's a girl!", only it isn't. Either you get into trouble for doing inappropriate things and wearing inappropriate clothes for your sex, or you live happily for a while, blissfully unaware that anything's askew. Then puberty hits. You may be depressed, even suicidal, and might self mutilate every now and then. Eventually you'll get yourself sorted and find a psychiatrist who can help you and give you hormones. Then you transition, living as your real gender, and eventually look like your real gender too. This can take a while and/or cost a lot. At this point your family may disown you and your friends may ostracize you, but don't worry. If they do, you'll get better ones who like you for who you are, not who you pretend to be. After a year of living in your chosen gender role, AKA the real life test, you can have sex reassignment surgery (If you want to; many transsexuals now opt to be non-op for various reasons) for a mere ten thousand pounds (If you live in the UK, that is. It's a lot of money, anyway). If you're lucky you might even fall in love with someone, but if they don't run away screaming or hurt you when you tell them your past, you can't marry them. Legally, you can never be male or female again if you want to do something like get married. (Update: this is now thankfully out of date, at least in the UK and Australia.)
OK, there seems to be a bit of confusion over the meaning of the word transsexual, so I'll try and clear it up. Please message me if I've made any mistakes.
For subjects people don't really feel comfortable talking about, they seem to pick Latin words to describe them. So you get phrases like trans (crossed) sexual (something to do with sex). Both of these words are a bit ambiguous. "Crossed" doesn't mean you cross from male to female, or from female to male; it means your gender is one and your physical sex is the other, hence they are crossed. "Sexual" causes even more confusion. It means something to do with sex, but in English sex can mean several things. In this context, it doesn't mean the action of sex, it means your gender or your body's physical sex. So while it might sound like the word could mean "someone who gets aroused by changing from male to female," it actually means "someone whose gender and sex are different from each other." Hence it has nothing to do with sexuality in the sense of what turns you on, even though it has the phrase "sexual" in it.