Prospex are seiko's more rugged line of watches for use when out in the field such as diving, hiking etc. They usually have screw down crowns, often ISO certified divers and can really take a beating. They are mid tier in the seiko lineup, usually priced between 500 and 1000 euros.
PADI is a subset of this, they are watches designed in collaboration with the Professional Association of Diving Instructors (PADI). They are mostly variations of Proxpex watches, usually with a pepsi style bezel insert.