A rinse and spin cycle on your washing machine means that the machine will simply rinse your clothes with clean water and then spin them dry. It's a quick cycle that requires no detergent and runs automatically after most other wash cycles.
You can also run a rinse and spin cycle on its own, without having to wait through an entire wash cycle, to clean any light blemishes on your clothes. Running an extra rinse and spin cycle at the end of a normal wash cycle will help ensure that all the detergent, bleach and other cleaning agents are completely removed. If you have allergies or sensitive skin that's irritated by detergent residue, performing an extra rinse and spin cycle could help reduce that irritation.