Attachment theory is a concept in developmental psychology that concerns the importance of "attachment" in regards to personal development. Specifically, it makes the claim that the ability for an individual to form an emotional and physical "attachment" to another person gives a sense of stability and security necessary to take risks, branch out, and grow and develop as a personality. Naturally, attachment theory is a broad idea with many expressions, and the best understanding of it can be had by looking at several of those expressions in turn.
Psychologist John Bowlby was the first to coin the term. His work in the late 60s established the precedent that childhood development depended heavily upon a child's ability to form a strong relationship with "at least one primary caregiver". Generally speaking, this is one of the parents.
Bowlby's studies in childhood development and "temperament" led him to the conclusion that a strong attachment to a caregiver provides a necessary sense of security and foundation. Without such a relationship in place, Bowlby found that a great deal of developmental energy is expended in the search for stability and security. In general, those without such attachments are fearful and are less willing to seek out and learn from new experiences. By contrast, a child with a strong attachment to a parent knows that they have "back-up" so to speak, and thusly tend to be more adventurous and eager to have new experiences (which are vital to learning and development, obviously).
There is some basis in observational psychology here. The baby who is attached strongly to a caregiver has several of his or her most immediate needs met and accounted for. Consequently, they are able to spend a great deal more time observing and interacting with their environments. Thusly, their development is facilitated.
For Bowlby, the role of the parent as caregiver grows over time to meet the particular needs of the attached child. Early on, that role is to be attached to and provide constant support and security during the formative years. Later, that role is to be available as the child needs periodic help during their excursions into the outside world. 
Mary Ainsworth would develop many of the ideas set forth by Bowlby in her studies. In particular, she identified the existence of what she calls "attachment behavior", examples of behavior that are demonstrated by insecure children in hopes of establishing or re-establishing an attachment to a presently absent caregiver. Since this behavior occurs uniformly in children, it is a compelling argument for the existence of "innate" or instinctual behavior in the human animal.
The study worked by looking at a broad cross-section of children with varying degrees of attachment to their parents or caregivers from strong and healthy attachments to weak and tenuous bonds. The children were then separated from their caregivers and their responses were observed. The children with strong attachments were relatively calm, seeming to be secure in the belief that their caregivers would return shortly, whereas the children with weak attachments would cry and demonstrate great distress under they were restored to their parents.
Later in the same study, children were exposed to intentionally stressful situations, during which nearly all of them began to exhibit particular behaviors that were effective in attracting the attention of their caregivers – a keen example of attachment behavior. 
Hazan and Shaver
Early on, one of the primary limitations of attachment theory was that it had only really been studied in the context of young children. While studies of children are often instrumental in the field of developmental psychology, that field is ideally supposed to address the development of the entire human organism, including the stage of adulthood. In the 1980s, Cindy Hazan and Phillip Shaver were able to garner a lot of attention, then, when they turned attachment theory on adult relationships. 
In their studies, they looked at a number of couples, examining the nature of the attachments between them, and then observed how those couples reacted to various stressors and stimuli. In the case of adults, it would seem that a strong attachment is still quite important. For example, in cases where the adults had a weak attachment, there were feelings of inadequacy and a lack of intimacy on the part of both parties. When attachments were too strong, there were issues with co-dependency. The relationships functioned best when both parties managed to balance intimacy with independence. Much as is the case with developing children, the ideal situation seemed to be an attachment that functioned as a secure base from which to reach out and gain experience in the world.
Criticisms of Attachment Theory
One of the most common criticisms of attachment theory is that non-Western societies tend to offer up compelling counter-examples. For instance, in Papua New Guinea or Uganda, the idea of a child being intimately attached to a caregiver is somewhat alien, and child-rearing duties are more evenly distributed among a broader group of people. Still, "well-adjusted" members of society are produced, indicating that, at least in these societies, some other mechanism is acting in the place of the attachments that are so necessary for Western children.
- Attachment theory states that a strong emotional and physical attachment to at least one primary caregiver is critical to personal development.
- John Bowlby first coined the term as a result of his studies involving the developmental psychology of children from various backgrounds.
- Mary Ainsworth conducted this research, discovering the existence of "attachment behavior" – behavior manifested for the purpose of creating attachment during times when a child feels confused or stressed.
- Hazan and Shaver in the 1980s demonstrated the applicability of attachment theory to adult romantic relationships.
- Attachment theory has had a profound influence upon child care policies, as well as principles of basic clinical practice for children.
- Critics of attachment theory point out the lack of parental attachment in many non-Western societies.
1. Bowlby, John. Attachment and Loss. 1969.
2. Ainsworth, M. “Infancy in Uganda: Infant Care and the Growth of Love.” Baltimore: John Hopkins University Press, 1967.
3. Hazan, C. & Shaver, P. “Attachment as an organizational framework for research on close relationship.” Psychological Inquiry. 5 1-22, 1994.