Chemistry of toilet bowl mineral stains on porcelain
Many places have water with dissolved iron and manganese, which precipitates out as a brown or black stain when exposed to air. 

Massachusetts has areas of podzol, where the fine sandy glacial till had been covered by pine forests. This makes for acid soil, which in anaerobic conditions readily dissolves iron in the ferrous form, $\ce{Fe++}$ from the ground, which is a pale green color, almost unnoticeable. If you dig into a podzol, exposing it to air, the soil turns brownish or orange in a matter of minutes as the iron is oxidized to its ferric, $\ce{Fe+++}$, form. That's what is happening on your porcelain fixtures.

It's hard to dissolve the brown ferric iron stain, but there are two ways to do so:


Reduce the iron (and manganese) to the lower-valence state. Sodium metabisulfite $\ce{Na2S2O5}$ is sold for this purpose under names such as Red-B-Gone. It works well on clothing and light stains on porcelain. Use this with caution, since it can generate noxious sulfur dioxide fumes if it comes in contact with an acid.
Dissolve the oxides in a strong acid, such as $\ce{HCl}$, hydrochloric acid. Though more dangerous than reducing agents, 10%-30% $\ce{HCl}$ can remove thicker layers of stain. Use with great caution, including eye protection and rubber gloves. Rinse immediately with plenty of water if any gets on you or your clothing! $\ce{HCl}$ is available in hardware stores and in plumbing supply shops.


You can also treat the water to remove iron and manganese or to sequester it with a polyphosphate additive.

Do not use alkaline drain cleaners (e.g. $\ce{NaOH}$ and Drano®) on vitreous china, as it etches the glaze and allows stains to soak into the porcelain, making the stains difficult or impossible to remove.