A purpose trust is a trust which is set up for the benefit of a purpose as opposed to the benefit of a person. When we hear of a trust in everyday terms, we normally hear of it in the context of a trust being set up for a person not a purpose — for example, we normally hear that John has set up a trust fund for his son. In contrast with this, a purpose trust is setup for a purpose, not a person — for example, a trust setup for the promotion of the principles of the free press would be a trust setup for a purpose.
A private purpose trust is just a purpose trust as I have explained above, setup by a private person. They are generally unenforceable.
A Charitable Trust is a particular type of purpose trust — a charity exists for a purpose. They are regulated by the Charities Acts.