NAUI has a few courses that are hard to compare with the other agencies. Their advanced rescue is like that. So is their master diver course. The NAUI program is flexible and works around these two courses, not making them specific requirements for any particular program. But they are there, and they contain extensive features that you do not see anywhere else.
NAUI's basic rescue and PADI's basic rescue, as well as that of YMCA Scuba and SSI, are fairly similar, in my opinion. These basic rescue courses are the mainstream rescue courses that are available in most locations. They are valuable courses to take because they make you a better buddy and also a safer diver yourself.
In the basic rescue courses of each agency, you learn about stress and its role in panic. You learn to observe stress in yourself and in others, like your buddy and other divers on the boat. You learn to deal with stress and defuse it.
You go on to learn about self-rescue to extricate yourself from difficult situations, and to assist other divers verbally from a short distance or close up, when they get into difficulty.
Then you go through various steps of a full blown rescue of another diver. The course usually goes over finding a missing diver. You do a grid pattern for a submerged marker or bright weight belt or something like that.
You would also practice following someone else's bubble stream downwards to locate them underwater.
Then you surface the victim, who is feigning unconsciousness, by either ditching their weight belt underwater or orally inflating their B/C a little. Various instructors teach this differently and there are several schools of though.
Once you have them on the surface, you ditch their weight belt if you have not already, and ditch yours as well, and remove their face mask. That should make them fairly buoyant, but if not, you would inflate their B/C as well. While you are doing all this you are supposed to call for help.
Next you check to see if the victim is breathing, and if not then you start rescue breathing (aka artificial respiration) on the surface. Now the in-water rescue breathing (aka artificial respiration) begins on the surface. Its good to have already taken the CPR First Aid class before you take a rescue class.
Then you begin to tow them back to shore in a dosido position. As you get close to shore, you need to ditch your own B/C and then theirs as well while you continue your rescue breathing. You practice that in the class on several rescue exercises. When you can stand near the shore, you ditch your fins, and your mask as well, if you have not already done so. When to ditch the mask depends on various instructors and the sea state. There are several schools of thought.
During the training, you normally have two students helping you carry all your gear and the victim's gear that you ditched. Hopefully you will not lose any gear during the training, but a lot of gear often gets strewn along the bottom during a rescue class. Its a good idea to have your name and phone number on your gear, in case you lose some of it. Divers are by and large pretty honest, and will call you if they found something of yours.
Once you are back at shore, there are about four ways to carry an unconscious diver up the beach, or to load him/her back onto a boat or pier. You will practice them. This is the hardest most exerting part of any rescue course. You will carry at least one person about your size up the beach to the dry sand. Then you gently lay the person down, scoop out sand under their head so that their head tilts back, remove their hood if they are wearing one, and check for a pulse.
If there is no pulse, you begin CPR combined with continued rescue breathing.
If there is a pulse, you continue rescue breathing, and instruct shore people to summon medical help. You continue your CPR and rescue breathing until the medical team arrives, even if you believe it to be hopeless. You do not stop.
Once the medical team arrives, you provide them with a complete report of what happened. Normally the sheriff's dept will show up as well and start to retrieve all the gear, and you will need to reclaim your gear, and identify the victim's gear. That is called accident management. There is also crowd control, which you would have to deal with before the medics arrive. You go over all that in the basic rescue class.
PADI teaches a well structured basic rescue class. Anywhere in the world that you take PADI's class, it will be fairly rigorous and cover almost exactly the same materials, depending on your instrutor a little. PADI programs and carefully structured and not designed to be very flexible.
Your NAUI course will depend a lot on the individual instructor. NAUI publishes general and specific standards, and then requires the instructors to 'customize' a course specific to the students and the given location. That is how the two courses would differ. But by and large they would be fairly similar courses at the basic rescue level.
Talk to the specific instructors who will be offering each course, and see how the chemistry works out between yourself and the instructor. I told you what a typical course would cover. Ask what your instructor will be teaching.
BEWARE of dive shop personnel who brush you off with 'oh someone will teach the class, it all depends, we cant tell you who the instructor will be.' Don't be taken in by a line like that. Find another dive store.