"Foreclosed identity" is the term used for a person who has ended up committing to a particular value system before they have really had a chance to discover what value system would fit them better. An individual might want or even try to explore, but because of the pressure placed on them by their culture or family they are "squeezed" in to a certain way of thinking. One family might tell their child that the family way of thinking is the only "true" and "right" way of looking at the world, and thus the individual comes to share the same set of values, even if they do not necessarily "fit" those values.
Let me give you two examples. A child is born into a fundamentalist Christian family who believes that homosexuality is "wrong". Even though this child may be homosexual, he or she will grow up thinking that homosexuality is "wrong" because of their family upbringing. Another example might be a child who has two doctors for parents, who pressurise (overtly or subconsciously) their child to follow in their footsteps and become a doctor. Even if the child is more suited to study and follow a career in the arts, for example, he or she may well "cave in" to pressure and become a doctor.
Having a foreclosed identity thus means that there is a huge discrepancy between your believed values and your genetic pre-dispositions. This discrepancy results in a huge effort to remain true to your believed values while your body and mind often leans towards following your genetic pre-dispositions. This can create stress and even illness.
Identity foreclosure is one step in the process of finding a sense of self. It occurs when people think they know who they are, but they have not even explored their options yet.