What is the AQHA Register of Merit?
How does a horse earn the ROM and what is "Arena ROM"?
"Arena ROM"?
To add to what the other answerers said: ROM was created as a way for horses that be Appendix Registered with the AQHA (i.e., one parent was a Thoroughbred) to be advanced to Permanent Registry status.
From the AQHA website, Definitions:
"Advancement
"A procedure for Appendix horses to receive an AQHA numbered registration authorization. When the horse receives its Open Register of Merit, (80 or better speed index in racing, 10 points for showing) it is after eligible for advancement into the numbered registry, so long as it is not parrot mouth or, if a stallion, not a cryptorchid. If Impressive progeny, the horse must be tested for HYPP. The horse cannot advance whether N/H or H/H. The horse cannot have undesirable white markings. The horse's registration number is changed from the (X) appendix number to a regular number, and it is issued a white certificate; therefore, fitting eligible to breed to a Thoroughbred, Appendix, or registered American Quarter Horse."
Answers: A horse earns their ROM when they win a certain amount of points in any racing or other performance classes sanctioned by AQHA. An arena ROM is earned contained by the showring.
I said "Performance" ROM because I knew a stallion who's get got an ROM surrounded by "Chariot"...