It means Private. In some baptisms, any of the following may be written in the margin or squeezed into the date box of a printed baptism register: P, private, priv
A private baptism was a baptism performed at the person's home, usually by someone other than the local priest, sometimes when the child's soul was feared to be in peril. For example, a midwife or schoolmaster or neighbor might baptize an newborn infant who was weak, sickly, and likely to die soon - they wanted to baptize the child before it died to ensure that it went directly to heaven.

Some members of the higher social class also had their children baptized at home, presumably to avoid mixing with the riff-raff at the local church.

Many private baptisms were never entered in the parish register; they were entered only if the family informed the priest and/or when the child was well enough to attend church and be "received into the church", at which point they received a conditional (official) baptism and the private baptism was noted in the register.

Also see "What does the letter "P" mean in a burial?"