Statuses
Statuses are workspace-wide and admin-editable, under Settings, Statuses. Each has a name and a colour, and the order you put them in is the order the board draws its columns in.
Most teams need three or four. A status nobody moves work out of is a status that should be a field or a filter instead.
Where a status shows up
In the table it is a filled plate in the Status column, in the board it is the column itself, and everywhere else it is a small dot in the status colour. Colour is never the only thing carrying the meaning, so a status is still readable if the colour is hard to distinguish.
Completed is a status like any other. Completing a task is done from the Status column or the card, not from the row's selection checkbox.
Custom fields
Admins define custom fields for the workspace and every task can carry them. A field becomes a column in the table, keyed to the field itself, so the width you give it follows it into every project rather than being set again per project.
Add fields when you find yourself putting the same information in task titles. Priority, client, effort and area are the usual four. Beyond that, most teams are describing something a project or a status would express better.