Salinity has got to be the most important, for the simple reason with out it everything will die. You need to aim for a salinity of 1.025. I would recommend using a Refractometer. I would try and avoid using a swing arm hydrometer, these are inaccurate. The Refractometer will give really accurate results and compensate the reading with the temperature of the water, they are really simple to use. They can be a little pricey but they are worth every penny.
Ammonia, Nitrite and nitrate. Check these in this order. You will really only need to check for Ammonia and Nitrite on a newly established Aquarium. These will go to zero after the Nitrogen Cycle. Then you really only need to check for Nitrate. Nitrate is the end product of the Nitrogen cycle. It is recommended that Nitrates need to be kept below 0.5ppm although many reef inhabitants will tolerate a little more. The easiest way to keep nitrates low is keep up with routine water changes. It is not uncommon to be able to run a mature aquarium with zero nitrates.
Phosphates. If you have high phosphates present living corals will die. Small amounts will be tolerated by most corals. This is one of the easiest unwanted chemicals to get rid of. All that is needed is a product such as Rowaphos. This can be placed in a filter sock and hung in direct flow in your tank or sump. Although it is expensive it is very effective. This media will absorb the phosphates until it is full. Then it will need replacing. Checks for phosphates must be done with other routine water checks.
Magnesium, alkalinity, and calcium. I have put these together because they need to be checked in this order as they affect each other, and if not tested and rectified in this order you will cause big problems.
Magnesium helps keep the balance of Alkalinity and Calcium, without it you would find it very difficult to maintain Alkalinity and Calcium at the required level. You should aim to keep Magnesium at between 1300 and 1500ppm. Many Hobbyist aim for about 1350ppm, this is about the amount found in natural sea water.
Alkalinity or Carbonate hardness needs to be maintained between 7dKH and 10dKH. I would recommend aiming for at least 8 though. This can be increased using a buffer.
Calcium is needed to help Hard Corals build their skeleton, you should aim for 400ppm. Without this your Hard Corals cannot create a skeleton, like us we need calcium to give us strong bones and teeth. The same applies for Hard Corals and many Invertebrates in our Reef Aquariums. Additives can be used to increase Calcium.
PH. Many Hobbyist recommend checking this, but i have heard many hobbyists don't. PH changes constantly and is affected by Light, Oxygen, Magnesium, Alkalinity and Calcium, and so results can be different every time. If you are going to check PH, check it at the same time of day to get consistent results. All other parameters should be checked in the order above leaving PH until last.
A word on dosing. If your parameters are out and need adjusting, don't panic. I recommend adding additives in the order given in this article, but do it very slowly. Don't rush, read the instructions on your additives. If you go too far just do a water change and check. If it still still high carry out further water changes. Remember the Corals, Fish and Invertebrates are sensitive animals and quick intensive changes will destroy them.
There are many other parameters that can be checked and dosed, but these can get very confusing and difficult to check for. I will continue these Articles on my blog covering more of these parameters including Strontium, Iodine, Iodide and any other i am able to research.