A non-drying oil is an oil which does not harden and remains liquid when it is exposed to air.
This is as opposed to a drying oil, which hardens (through polymerization) completely, or a semi-drying oil, which partially hardens.
Oils with an iodine number of less than 115 are considered non-drying.
Non-drying oil is often used as a base in anti-climb paint, a type of slippery coating used to prevent climbing on its surface.
[1] Another use would be in baby oil.