Transgender people are people whose gender identity is different from the gender they were thought to be at birth. "Trans" is often used as shorthand for transgender.
When we're born, a doctor usually says that we're male or female based on what our bodies look like. Most people who were labeled male at birth turn out to actually identify as men, and most people who were labeled female at birth grow up to be women. But some people's gender identity, their innate knowledge of who they are, is different from what was initially expected when they were born. Most of these people describe themselves as 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 as a combination of male and female. 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 or genderqueer.
Everyone, transgender or not, has a gender identity. Most people never think about what their gender identity is because it matches their sex at birth.
Being transgender means different things to different people. Like a lot of other aspects of who people are, like race or religion, there's no one way to be transgender, and no one way for transgender people to look or feel about themselves. The best way to understand what being transgender is like is to talk with transgender people and listen to their stories.