R&I: This is an acronym for Remove and Install. Refers to a part removed from a damaged vehicle to be saved and reinstalled after the repair has been completed. In many cases, to repair damage to the outside of a vehicle, interior trim, seating, etc. must be removed to make a proper repair.
R&R: This is an acronym for Remove and Replace. It refers to a part removed from a damaged vehicle that cannot be acceptably repaired. It is replaced with a new part. 
Rebuilt Part: May be either an OE or imitation part. Generally rebuilding consists of replacing only those components that may be broken or unusable and may not involve a complete disassembly and inspection. They typically come with a limited warranty (30 to 90 days). Rebuilt parts are typically supplied by an aftermarket source.
For more information on Rebuilt Parts.
Recycled Parts: Also referred to as salvage, used, reconditioned or rebuilt parts. They are sometimes obtained from junkyards and can be either OEM or aftermarket parts.
For more information on recycled parts.
Remanufactured: Remanufactured generally means parts removed from an existing vehicle that are repaired and/or refinished. Honda Genuine Remanufactured parts begin with a used part that is completely disassembled, inspected, diagnosed, cleaned and any worn or inoperative parts are replaced. The part is reassembled and tested to ensure the part meets the same specifications as the original part. Honda Genuine Remanufactured parts are literally “as good as new” and are backed by a warranty honored by Honda dealers nationwide.
For more information on remanufactured parts.
Repair Authorization: The point at which a consumer authorizes the repair to their vehicle (and in some cases contingent upon the Insurance Company settlement process). 
Replacement Cost: The cost to repair or replace an insured item at the present time, according to its current worth. 
RO: This is an acronym for repair order, also called a work order. This is the document which will be used by the body shop to keep track of the time spent, expendable materials consumed (such as paints, etc.) and parts used to repair a collision damaged vehicle.
Rubbing Compound: An abrasive paste used to smooth and polish paint films. This is also commonly known as polishing compound.