What is a psychoanalytic approach?

Psychoanalytic psychotherapy is primarily concerned with the internal world of the patient.
This means a focus on what goes on in the mind of the individual in relation to experiences built up since infancy and continuing into adulthood. Early experiences with significant others, especially parents/carers and siblings, result in particular ways of being in the world, as we adapt to meet the developmental and external challenges we each face growing up.
Psychoanalysis as originated by Sigmund Freud, and developed by subsequent and contemporary psychoanalysts, provides us, for example, with the concept of the unconscious. The idea of the existence of an area of the mind outside of daily conscious control, but nevertheless significantly influencing our behaviour, can still be seen as a radical idea. In psychoanalytic psychotherapy the unconscious is seen as the central arena within which many internal struggles between past and present are faced. For example, someone may have a deadened attitude to life where they struggle to find meaning. That person may want to change and obtain some pleasure from life, but s/he may feel unable to let go of a more familiar psychological position.
To understand more about this model of psychotherapy there is a helpful article on the British Psychoanalytic Council website entitled Making Sense of Psychotherapy and Psychoanalysis, which you may wish to read.