I have read in many places that transgender people are usually people who are born with typical male or female anatomies but feel as though they've been born into the “wrong bodies.” The thing that I am confused about is that how someone can not be considered as normal just because he thinks he should not be what he is? I met a female who looked exactly like a normal female and she was very beautiful. She didn't know that she was a transgender till her mom told her that she was transgender when she became adult. What is biological difference between a transgender and a normal male/female?
Remi gives a good introduction to the various terms and definitions you should know when learning about gender identity. However, he didn't really address your basic question:
What is biological difference between transgender and a normal male/female[?]
Except for some likely genetic differences, which we aren't even close to beginning to understand, there is no biological difference between two persons born with $XX$ sex chromosomes, one of whom identifies as female (cis-gender) and one of whom identifies as male, or somewhere on the spectrum between female and male (trans-gender). There is no difference physically - both have the same organs, including sex organs, and both have the same secondary sex characteristics. One cannot tell the difference upon gross physical examination, X-ray, blood test, etc. Gender is a social construction, and while the majority of people in the West have a gender identity that matches their biological sex, some do not, for reasons that are not yet clear.
Let's first consider the difference between the concepts of sex and gender.
The distinction between sex and gender differentiates sex (the anatomy of an individual's reproductive system, and secondary sex characteristics) from gender, which can refer to either social roles based on the sex of the person (gender role) or personal identification of one's own gender based on an internal awareness (gender identity).
Below are some definitions of the term you used (transgender) and of terms that some people confuse.
A transgender person is someone who have a gender identity that differs from its sex. For example, a man (with standard male organs) who identifies as a woman.
Because (1) sexual and romantic attraction are often important driver of gender identity and because (2) many people confuse the terms relating to sexual attraction with the previous ones, I wanted to add the definitions of
A heterosexual is someone who is attracted to people of the opposite gender.
A homosexual is someone who is attracted to people of the same gender.
A bisexual is someone who is attracted to people of both genders.
While all of the above definitions were referring to social gender identity (transgender), sexual and romantic attraction (homosexuality, heterosexuality, bisexuality) and medical intervention (transexual), in my list of terms, only the term intersex actually refers to a 'biological' difference.
You should read the linked wikipedia articles who will teach you much more about the definitions and slight variations behind these terms than I could do in the such short definitions I used. There other other concepts you might want to investigate the definition such as gender non-conformity, genderqueer, third gender and pangender for example.
You will note that as only one term was referring to 'biological' variation, the question could be answered in much more depth by a specialist of gender studies (which I am not) rather than by a biologist (which I am).
Finally, note that this post is closely related.