We have two main types of corrector pens - ink eradicators, that erase washable fountain pen ink, as well as ink from certain other pens; and correction pens that write a line of correction fluid, like Tipp-Ex.
Ink eradicators only work with certain types of ink. For fountain pens, these are often referred to as washable or eradicable, and are most often available in blue. Many ink eradicators have a blue tip at the other end, for rewriting, as the original pen would struggle to write where the eradicator has been used! Washable inks include Parker Quink in Washable Blue, or Pelikan 4001 ink in Royal Blue. Other pens designed to work with ink eradicators include the Stabilo Easy Original and the Pilot V-Pen Erasable.
Correction pens are a more convenient way of putting down correction fluid, and can be more accurate than a bottle of Tipp-Ex or Liquid Paper. While on the subject of Liquid Paper, it is, of course, mandatory to mention that it was invented by Bette Nesmith Graham, mother of Monkee Michael Nesmith. They apply a thin layer of opaque white paint over the error, which should be almost invisible on white paper.