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The most common tests for aquarium water are, Ph, Ammonia, and Nitrite.
Of the above 3, Ph is the least important, but yet it is the most familiar to people, so if a new aquarium keeper gets a test kit on his own decision, it will likely be a Ph test kit, and he will wonder why his fish keep dying.
I like the Marine Enterprise test kits best for testing Ammonia, and Nitrite.
When testing for Ammonia, it should be les than .025 ppm. If it is higher, It needs to come down. If it is higher than .1, then you need to do a water change.
When testing for nitrite, it should be les than .1 ppm. If it is higher, It needs to come down. If it is higher than .25, then you need to do a water change.
Nitrate is less toxic, but if it gets higher than 50 ppm, then it is toxic to the inhabitants. Ideally it should be 10 ppm or less.
See Nitrogen Cycle.