Intersex: "Sex" is different from "gender." "Sex" refers to biological and physiological characteristics, like genital organs, hormones the body produces, and chromosomes a body has. If someone is intersex, they were born with those characteristics not completely male or female. According to the Intersex Society of North America, there's no clear agreement on exactly how to define intersex, but some statistics show it might be far more common than people imagine. For example, babies born with chromosomes that are not XX and not XY, or people having surgery to "normalize" genital appearance — each occur more often than once in 2,000 births.
Transgender: "Gender" is more outwardly constructed, and refers to the attitudes, feelings and behaviors that a culture associates with a particular biological sex. If someone is transgender, they learn that the gender they feel they are is different than the one they were assigned by other people. They may or may not get surgery to align the gender they realize they are with the physical characteristics associated with the gender.