A spinner is a streamlined fairing fitted at the centre of a turbofan engine. Spinners both make the aircraft overall more streamlined, reducing aerodynamic drag and also smooth the airflow so that it enters the air intakes more efficiently

The reason for painting designs on engine spinners is for the safety of ground personnel. Working near a running jet engine is extraordinarily dangerous. An engine, even at idle thrust could suck a human that walks in the hazard area in to the engine. It is absolutely critical that ground crews can identify a running engine and stay away from it.

Also notice the Semicircular slots in the last photo, these are for a tool to remove the spinner from the fan disk.