Whenever someone was turned whatever made them special as a human was heightened. For Carlisle it was his compassion. For Esme it was her ability to love completely. For Emmett it was his strength. For Rosalie it was her beauty and grace. Now for the special Cullens: Jasper was very charismatic, he could manipulate people easliy with his charm. When he changed he had the ability to feel and control the emotions of all those around him. Alice was considered crazy for the things she saw as a human and was put in an asylum. When she was changed, the things she saw became more clear and easier for her to understand. Now for Edward. You see as a human Edward was a very good listener and he was very empathetic towards people(he had a good understanding about people he met) So when he was changed he was able to read peoples thoughts.

in twilight the book or movie, yes he is the only one, but in new moon you're introduced to the volturi. aro one of their leaders and one of the oldest vampires in the world can hear every single thought a person or vampire has ever had, but he has to touch them first.
edward can hear the thoughts from far away, but he can only he what they are currently thinking.