This stems from the work of Carl Jung. The central idea is that what we do and feel, and how we think of ourselves and other people, depends upon forces and processes we are not aware of - the 'unconscious'. These may be part of our common human nature or particular to the individual.
The analytical psychologist (also known as a Jungian analyst) tries to understand these unconscious elements in what the patient says, and the dreams they report. Particular attention is given to how the patient regards the analyst and vice versa. It can be beneficial for a patient to understand how this 'transference' mirrors the early family situation, enabling them to take conscious control. Jungian analysts believe that we are all highly capable of healing ourselves and taking charge of our lives.
Analytical psychologists treat people suffering from emotional disorders or problems with relationships, but can also help 'normal' people who are discontented with themselves or have lost a sense of direction in their lives. So it is not necessary to be 'ill' to benefit from Jungian analysis, since self-discovery and the exploration of the inner world are valuable in themselves.
Analytical psychology also has much to say about society and culture, showing how unconscious processes affect the groups and institutions we belong to.