Marine tanks fall into two main categories. Fish and corals.
If you have a fish only tank, you monitor ammonia, nitrite, nitrate, pH, salinity (salt level) and temperature.
If you have a coral tank, then you monitor these as well as calcium, GH, & KH. Calcium is more important for hard corals like staghorns, brains and other corals that make a hard calcium based skeleton. Calcium is less of an issue if you keep soft corals like leather corals.
You can have some fish in coral tanks but generally you don't have many fish in coral tanks. Coral tanks are set up specifically for corals and a few fish are added after to pick up algae and add some movement.
Fish only tanks can have fish, live rock and shrimp or crabs if you want them. A lot of people go half and half and have a few corals in with their fish, and they add things like starfish, sea urchins, shrimp and hermit crabs.
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If you do a regular water change (every 2-4 weeks) on the marine tank, that is pretty much it. However, if you don't want to do water changes on a coral tank, then you have to monitor the minerals and trace elements and add supplements regularly to keep everything stable.
In a coral tank, there is usually a lot of live rock. This is simply rock that has been in the water for 6 month or longer. If you have decent sized pieces of live rock that is limestone or sandstone, you get anaerobic bacteria inside the rock and these break down nitrates and remove them from the water. A lot of people also have a sump/ refugium under the main tank and they grow marine algae in the sump. The marine algae use up nutrients like ammonia, nitrite, nitrate and phosphate.
On the outside of the live rock is aerobic bacteria that convert ammonia into nitrite and nitrate. The live rock can actually do the job of the filter when it comes to removing ammonia and nitrite. However, live rock does not get rid of fish food or waste from the aquarium and you need some form of mechanical filtration for this. A normal aquarium filter with some sponge in will do the job.
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A lot of marine tanks have protein skimmers on/ in them and these help to remove protein from the water and keep the water in better shape for longer. Less protein means less ammonia and subsequently less nitrite and nitrate. There are numerous different models and some are better than others. They remove protein but many will also remove plankton from the water and this can be an issue if you are keeping live corals because many of the corals like to catch plankton to eat. You can supplement plankton with liquid invertebrate food or newly hatched brineshrimp.
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When I had marine tanks I did a 90-95% water change once a month using natural sea water from the beach. I monitored the salinity and didn't really bother about much else. The big water change each month removed any nutrients that were building up and replaced all the minerals and trace elements the corals had used.
I had marine algae (Caulerpa species) in the tanks and these sucked up nutrients, and I didn't have many fish in each tank.