It's been awhile since I have disagreed in part with a lot of the posts on a thread. Here is my take on water testing and quality.
First, before you even bother to test anything else, make sure your SG is correct. If it's not, most of the other tests will be skewed and of little value.
Ammonia and nitrite - It's necessary to test these parameters during the initial cycle of a tank. Once you get past that, unless you have something really really wrong, you seldom need to test for them. A reef system has a huge capacity to convert ammonia --> nitrite --> nitrate.
pH, calcium, alkalinity and to a lessor extent magnesium are critical to maintaining corals, so you should test for these also if you have any corals. It's usually best to test your newly mixed SW to establish a base line. Then when you test your tank you can spot any differences. You might want to add a supplement to you new SW, rather than the tank, or even change salt brands if you fund your current brand mixing to different levels, or being really low in some areas.
Nitrate and Phosphate - In a reef system these measure roughly how much pollution you have in the tank and how likely it is that you'll have algae issues. It's best to keep them as low as possible.
To keep things interesting, don't think for a moment that because all your test kit reading are good that you don't have a water quality issue. Algae can use up a lot of nitrate and phosphate giving you a massive growth of nuisance algae, and very low readings for nitrate and phosphate. That's one example only. There are plenty of others.
Above all, observe your livestock. If it's doing well, be extremely careful about making changes. Don't ruin a good thing. Livestock is often a better indicator toward water quality than almost any test you can run.
Also, very important, don't chase numbers. If something is way out adjust it slowly. It's really easy to mess up a whole system because a bunch of additives were added that were used too quickly or not necessary.
Lastly while you can debate "water quality" verses "water chemistry" they are so related that you really can't separate them. To keep corals all of the parameters need to be within reason. There are also plenty of other parameters you can test for. Most of them are only worth doing in specific cases. Of course, the vendors are always happy to sell you an additional test kit. There are also plenty of parameters we can't test for. So despite what some may say, keep you with your water changes.