What does calcium hardness, CO2, etc. have to do with water quality?

Jonathan asked:


What does calcium hardness, CO2, PH, nitrate, and phosphate have to do with water quality? Would you want these levels to be higher or lower if you wanted good water quality? What’s a generally ‘good’ level of these for water?

Would your answers be different if we were discussing quality in terms of ecological standards or human consumption standards?

You don’t have to answer every one of these questions, just a few answers will be well-appreciated. Thank you.