This is a concept I have recently come into contact with (at the time of writing: 2009), and in many ways it is a little bit bizarre. The idea is that it is possible to have multiple independent personalities which we switch between in particular circumstances. Not everybody is going to have this, but it apparently more common than one would think.
In its extreme form, it is a condition known as Dissociative Identity Disorder. When it is extreme like this, and the person is operating as one personality, they often don't remember things they did in the other and vice versa. When a person keeps switching from one to the other in rapid succession, it is difficult for other people to cope with, and they will generally get referred to the men in white coats. A classic example of this is in the Lord of the Rings: The Two Towers. There is a scene where Gollum keeps switching back and forth between the evil ring-obsessed Gollum and the more timid Sméagol.
It is essentially a defence mechanism which has its origins in some event or situation, often in childhood, where the person is unable to cope. It is created to shield the person from excessive pain in that situation and future similar situations. Whilst it may serve that purpose, the real problem is that such defence mechanisms remain in place even when the threat isn't really there. But it becomes more convenient, and also a habit, to let that personality take over so as to prevent the original pain from reoccurring.
The concept of a broken heart can be thought of as the fracturing of personality that takes place. Sometimes whole aspects of a person's personality can get masked and very rarely see the light of day. For example, you may have been carefree and fun-loving as a child, but then something happened that made you think that it was not safe to be like that, so a personality gets created that is serious, closely controlled and risk-averse so as to be safe, and that sticks.
The goal of the ministry into the fractured personality (also called parts or dual soul or alter egos or just alters) is to re-integrate the person by introducing the different parts to Jesus. Typically what happens is that one part is dealt with at a time, and the ministry receiver either becomes that personality or talks to that personality in the third person, relaying what it says to the minister. The minister then converses with that part and encourages it to have a two-way conversation with Jesus (through the Holy Spirit, i.e. for real, not imagined). If it's personality developed as a defence-mechanism, then the goal is that it should go to be with Jesus permanently. This then allows the real person that was hidden to become part of the whole person.
On the face of it, it does all sound a bit strange, and to be able to minister this, it does need a level of training. It's not something you can do on your own. Attempting to do this without any training will tend to leave the minister confused and discouraged, and the parts not dealt with. That's not to say it can't be learned by ordinary people, otherwise nobody would be able to do it.
Having parts can be a significant hindrance to all the other areas of ministry described in this section. For example it is possible to have a in church part, which receives all the areas of ministry really easily, and and everyday part, which hasn't received any ministry at all. Thus the ministry received has absolutely no effect on everyday life (e.g. at work), but in church the person is highly sanctified.
An example of this ministry is Bethel Sozo's Shabar.