The best known are transsexuals, officially termed 'Gender Identity Disorder'. I've recently met a teenage transsexual, a boy with a girl's body, in the volunteering program I'm in. He's a participant in the older group and a volunteer for the younger group. He's got a rough situation, but he's handling it very well - better than I would've. For more information on transsexuals, check out Transgender Care or one of the many other websites devoted to transgender issues.
What many people don't know is that transsexualism is only one kind of identity weirdness. There are many others as well.
A lot of people confuse identity differences with weird sexual kinks. Even transsexuals are sometimes confused with transvestites - people, usually men, who dress in the opposite gender's clothes for sexual excitement but are secure in their gender identity. For transsexuals, it's not a sexual thing, it's an identity thing. And that's true for all the conditions I'm describing here.
First, there are the transabled, also known as people with body integrity identity disorder - people who, though non-disabled, feel that their true self is someone with a disability. Most often they want to have a limb amputated, though spinal cord damage and blindness are common too, and some identify with other disabilities. If they can't get someone to surgically induce a disability, they might pretend they have it, such as a transabled paraplegic using a wheelchair even though his or her legs work fine. For some reason, a lot of disabled people seem to be really offended by the existence of these people, which makes no sense to me as a disabled person - just like the negative reaction to transgender people by many feminists makes no sense to me as a feminist. I don't think being disabled is a bad thing, so why would it bother me that someone wants to be disabled? For more information on transabled people, check out Transabled.org.
A condition some people confuse with being transabled is adult baby syndrome (adult child, teen baby, etc)also called transaged, age identity disorder or infantilism. These people identify as being a much younger age than they actually are, such as feeling they are in some sense a baby despite being physically adult. The thing many of these people hope for the most is to find someone willing to roleplay as a parent or babysitter and take care of them while they drink from a bottle, wear a diaper, and play with baby toys. Again, this is not a sexual thing - although there are 'diaper lovers' who are sexually attracted to wearing diapers, they are separate from adult babies in the same sense that transvestites are separate from transgender people. Many adult babies still act as adults for much of the time, but want freedom to act as a baby from time to time. For more information, check out Little AB's website.
Another category includes otherkin, furry lifestylers and others who feel they are not truly human. Inside, they feel like they are actually a different species, whether some sort of animal or anthropormic animal or a mythological species of some kind. This has also been described as 'species dysphoria'. This is one of the harder forms of this condition for me to understand, because they don't just have a human body, but a human mind, and in that sense will never be anything but human. For example, sometimes I'd like to be a cat, but I know my cat sees the world in a way I will never understand because I do not have a cat's mind. It makes more sense to me, actually, for people to identify as mythological creatures, because many of these creatures are depicted as having reasonably humanlike minds, such that a human could conceivably think like them. Nevertheless, this is something some people seriously identify as, and I respect that. I don't understand it, but I respect it. For more information on otherkin, check out Otherkin Alliance.
There are probably other identity conditions in this cluster that I'm not aware of. Most of these conditions have little, if any research into them, and so not much is known about what causes them. Likely if we figured that out, we'd know a lot more about identity than we currently do. The human brain is an amazing thing, and some very strange phenomenon can result from it. But at the same time, we must remember that these are not just phenomenon, they are people, and many of these identity conditions carry with them a great deal of suffering. It can be extremely distressing to feel like you're in the wrong body, no matter what you feel your true form is.