A popular image of transgender people is that of a "man trapped in a woman's body" and vice versa, but this isn't entirely accurate. A more accurate way to put it is that transgender people are people who are born into a body not associated with their gender, or were assigned a sex that does not match their gender.
This is a very large area and there isn't always agreement. One thing that is important to remember, though, is that most terms for trans people are based on gender. For instance, a man who was assigned female at birth would be called a trans man, a woman who was assigned male at birth would be called a trans woman. Many people who are new to transgender issues will trip up on this, but it's an important distinction.
There are two reasons for this. The first is the difference between gender and sex. A trans woman is a woman- she is in no way a man other than that she may have been raised with the expectation of growing to be one. Although she was assigned male at birth, this makes no comments on her gender.
Transgender People and Bodies Edit
The way transgender people feel about their bodies is extremely varied. Some people get highly offended at the idea that their pre-transition body is the right body because it belongs to them, while others get equally offended at the idea that there's something wrong with their body. Because of this it's important not to make assumptions about how trans people feel about their bodies, whether or not they plan to transition (or how far/in what way), and it's best to listen to each person's feelings.
One person wrote up a good explanation of how and why transgender people could feel that they do not have the wrong body.
Transgenderism and Sexuality Edit
Although many people have the idea that trans people are "just gay and trying to find acceptance", transgender people have just as vast a range of sexualities and romantic attractions as cis people.
Sexuality for trans people, particularly non-binaries can be quite complicated, though. Typical conversations about sexuality revolve around sex (assumed, assigned, or your current genitals) rather than gender, while many transgender people feel that basing their sexuality on their gender is more accurate. Some people prefer terms such as androphile (attraction to men and/or assigned-males) and gynephile (attraction to women and/or assigned-females) as they remove assumptions about your sex.
This can be easier for people who are gay,asexual, bisexual,polysexual, sapiosexual, or pansexual, as they don't make obvious assumptions about sex as gay/straight/lesbian do. It may also be considered easier to find people with these orientations who is fine with dating a trans person, but this is not always true.
Many binary-gendered trans people base their sexuality on their gender, though. For instance, a trans man who is attracted to men would be a gay man and one who's attracted to women would be straight. At the same time, a trans woman who's attracted to men would be straight while one who's attracted to women would be a lesbian. Although this may seem confusing, particularly if the person is non-op, the reason is that for the most part, a trans man wants to be a person's boyfriend and a trans woman wants to be a person's girlfriend. Even if they have to present as something else to the public, they still want their partner to see them as who they are and be in a relationship based on that.