The term transphobia is used to describe a prejudice, fear, hatred, and/or general negative feelings or beliefs held for people who are transsexual or transgender (regardless of how the person who is the target of hatred identifies themselves). Transphobia, like homophobia, isn't an actual clinical medical "phobia." No one is born transphobic. While fear and hatred may be feelings all people have, directing them toward a group of people based on their perceived gender identity is something we are taught. It's something we learn to do and something we can work to change.
Transphobia takes many forms. In it's most extreme (but all too common) form, people will physically assault and murder individuals who they identify as trans. Another example of transphobia is the belief that people who have transitioned from one gender to another, or who identify as gender variant, genderqueer, or trans, should not be in positions of authority (be it a teacher in a public school, the head of a community group, or the CEO of a bank). Another common example of transphobia is when people refuse to let trans people use the washrooms that match their gender identity.
Transphobic beliefs are deeply embedded in most societies and cultures, and they extend beyond fear and hatred to a general negative feeling toward people who are trans and the belief that they are inferior.
Read more about transphobia:
Trans Accessibility Project: Transphobia and Discrimination
Sylvia Rivera Law Project: Trans 101