There is a huge debate about inking. There are as many ways of inking as there are drawers, so I'm therefore going to restrict myself here to the principal ways of inking and to certain fundamental rules.
With regard to tools, anything that lets you put ink on your paper is allowed: quill, brush, biro, felt pen, etc. The most important thing is that you find the tool that suits you best, so don't hesitate to try them out.
Inking a drawing is "fixing" or finishing it. Generally, as below for example, we will ink a more or less raw pencil drawing. Inking enables the drawing to be re-done one last time, giving it a more contrasted, cleaner appearance. A long time ago, this was a stage that was almost obligatory to be sure that the drawing would correctly depict the impression. Today, this is no longer the case but the practice of inking remains widespread for historic and aesthetic reasons.
There are several ways to ink. The simplest comprises taking your drawing and then going over the pencil with ink, as in the example below. When the drawing is completely inked, you rub out the pencil, thus leaving only the ink and giving a "clean" drawing.
For various reasons, for a long time drawers have sought not to draw directly on the pencilled original. The technique most currently used for this is the light table. The principal is simple: you place a virgin sheet over your drawing and then put it all on a light tray so that the drawing appears on the clean sheet and you can thus re-do the drawing in ink without destroying the original.
If you don't have a light table, you can use chalk sheets directly. If not, digitalisation has changed things alot and some drawers will scan their drawings in order to print them in a primary colour (see drawing below) and ink them later. By rescanning the drawing it will be easy to make the background colours disappear and thus keep the inking itself.
If you have made a very clean pencil drawing, you can also then rework it in a programme for retouching images, in order to "imitate" the "inked" look, as in the following example.
There is no limit to the options for inking. Some will even ink directly to your tablet. It's up to you to test and find the method to suit you best.