The ANO's Article 130 is the definitive text on this.
Generally speaking, a private flight is one for which no valuable consideration is given or promised in respect of the flight. Note that this doesn't necessarily mean that just you, the pilot, can't take remuneration. No one can.
There are a number of permitted exceptions to this:
1) Hiring an aircraft: while the aircraft must be on a PT C of A, the flight is classed as private for other purposes.
2) Reimbursement of costs for flying displays, races
3) Charity flights with the permission of the CAA (general permission is laid out in an AIC)
4) Cost sharing when no more than 4 people are carried and the flight is not advertised outside a flying club [Note also that the pilot may not be employed as pilot by the aircraft operator -- you can't cost share if a club instructor is the pilot.]
5) Reimbursement of direct costs by employers of the pilot
6) Contributions by co-owners to group funds
For details, see the ANO (CAP393).