What is an over-current protective device?

That is called a fuse.

A fuse is a protective device, which will interrupt the flow of electricity if it is subjected to a current that is greater than its rating.

Some fuses are designed to handle a large inrush of current and are used in circuits such as air conditioners, where a large amount of current is needed at startup, for a short duration and then current drops to a lower level.

These are sometimes called time-lag fuses.