A private prescription is merely a prescription written by a doctor in private practice. Private prescriptions are not funded by the NHS. An NHS doctor may write you a private prescription for medication the NHS will not fund. Travel medication, for example, is not funded by the NHS, so you will receive a private prescription for something like malaria tablets.
Doctors usually charge patients for writing private prescriptions. The British Medical Association recommended in 2009 a fee of £15.50 per prescription for private practice. However, doctors are free to charge what they wish so it could be considerably more or less.
On top of paying your doctor the cost of writing a private prescription, you will also need to pay the cost of medicine dispensed on private prescriptions. This cost can vary greatly as, again, there is no fixed price for the medicine dispensed privately.