It has a few names. Person-centred therapy, person-centred counselling, PCT, client-centred therapy, etc.
However, if you see the term 'person-centred' it should mean one thing. It's all about you, how you feel and what life is like for YOU.
The idea of taking a 'person-centred' approach to therapy was put forward by Carl Rogers in the early 1940's. It was a brave move at the time because psychiatrists were following the Freud or Jung models which assume that the therapist is the expert within the therepeutic relationship, not the patient, or as they are more often referred to now, the client.
There are still forms of therapy where the client is told what to do or led in certain directions. With Cognitive Behavioural Therapy (CBT), for example, the client will be addressing a specific behaviour that may be causing them problems in life and the therapist may make suggestions as to how best improve their situation. Sometimes this can be like a coping mechanism or a temporary fix that will help the client but does not necessarily help with things that affect people on an emotional level.
The beauty of the person-centred approach is that it acknowledges that the client is the expert and the therapist is very much led by the client into the areas they feel they need to go. This can help the client feel more relaxed which can, in turn, allow trust to build faster between the client and therapist. As the trust strengthens then the client is prepared to dig deeper and deeper into their life and experiences, knowing that they are in control and can return to the surface whenever they want to, without undue pressure to explore areas that leave them feeling unsafe or uncomfortable.
If the therapeutic process were some sort of vehicle that is being maneuvered through a person's life then it is that person at the wheel. Sometimes the therapist may want to stop somewhere to invite the client to elaborate or explain in greater detail certain elements along the way or maybe even suggest the client considers something they just said. But the important thing for the client is to understand that they are free to decide where they are comfortable to go and what they are comfortable to talk about.
It should be made clear that this does not mean that therapy sessions should be use for the client to just come in each week to unload all the annoying stuff that has happened and tell stories from their life. In order to get the most from a series of sessions, the client should be prepared to look at some very sensitive areas of their life and try to analyse, with the help of the counsellor, what it means, how it affects them and how to move on from it. But, importantly, the person-centred approach mean that this can be done at the client's pace.
An additional benefit of this type of therapy is that because it focuses on the client's feeling and emotions, it is effective for a greater range of issues that affect people during their life.
If you are struggling with a certain type of behaviour then it's likely that you will benefit more by visiting someone that specialises in behavioural therapy. However, something that is affecting you on an emotional level requires a different skillset, and if you think about it, an awful lot of the things that can impact negatively on an individual's life will be on an emotional level. Stress, anxiety, bereavement, grief, trauma, abuse all affect the individual emotionally in some way. The problem that needs to be addressed is not specifically how the person reacts to the problem, but more about how they feel about it and the emotions that are brought up because of it.
With most things in life that affect your well-being, either positively or negatively, the affect will be on an emotional level, and it's at that level that person-centred therapy is most effective.