It depends on the state and age of your tank. In the beginning, you should test for:
Salinity/Specific Gravity
Ammonia
Nitrate
Nitrite
Phosphate
pH
Calcium
Magnesium
After you are comfortable with where your tank is at, you can back off the testing. I don't test anything in my tank unless I'm dosing something or have a specific problem.
One thing to NEVER do though, is to add anything to your tank (cal/mg/trace elements, etc) without testing for them. Too many people end up pouring crap into their tanks that the tank doesn't need. And they don't test for them, so they have no idea if they are in an appropriate range.