Transgender people are people whose internal knowledge of their gender is different from the gender people thought they were when they were born. “Trans” is a shorter word meaning transgender.
A transgender woman lives as a woman today but was thought to be male when she was born. A transgender man lives as a man today but was thought to be female when he was born.
Some transgender people identify as neither male nor female, or they may identify as both. There are a variety of terms that people who aren't entirely male or entirely female use to describe their gender identity, like non-binary, non-conforming, queer, and genderqueer.
For more information on transgender people and the trans movement, check out the National Center for Transgender Equality.