Emotional detachment, in psychology, can mean two different things. In the first meaning, it refers to an inability to connect with others on an emotional level, as well as a means of coping with anxiety by avoiding certain situations that trigger it; it is often described as "emotional numbing" or dissociation. In the second sense, it is a type of mental assertiveness that allows people to maintain their boundaries and psychic integrity when faced with the emotional demands of another person or group of persons.
First sense: inability to connect
Emotional detachment in the first sense above often arises from psychological trauma and is a component in many anxiety and stress disorders. The person, while physically present, moves elsewhere in the mind, and in a sense is "not entirely present", making them sometimes be seen as preoccupied or distracted.
Still, it is often not as outwardly obvious as other psychiatric symptoms; people with this problem often have emotional systems that are in overdrive. They have a hard time being a loving family member. They avoid activities, places, and people associated with any traumatic events they have experienced. The dissociation can also lead to lack of attention, and hence to memory problems, and in extreme cases, amnesia.
Second sense: mental assertiveness
Emotional detachment in the second sense above is a positive and deliberate mental attitude which avoids engaging the emotions of others. It is often applied to relatives and associates of people who are in some way emotionally overly demanding. A simple example might be a person who trains themselves to ignore the "pleading" food requests of a dieting spouse. A more widespread example could be the indifference parents develop towards their children's begging.
A more extreme form of this has been called "tough love," meaning letting someone go through a painful life experience without interference for the sake of its greater educational value. This can be an excruciating experience for loved ones, who must avoid the urge to step in and rescue the person from that pain (but thereby interfere with the loved one having a much-needed growing experience).
This detachment does not mean avoiding the feeling of empathy; it is actually more of an awareness of empathetic feelings that allows the person space needed to rationally choose whether or not to engage or be overwhelmed by such feelings.