What is the correct term for fill up or fill out?

The correct term for fill up or fill out depends on context and whether you are speaking formally. Both fill up and fill out are correct phrasal verbs but have slightly different uses. Fill up means to make completely full to capacity, such as fill up a gas tank, fill up a glass, or a room filling up with people, emphasizing fullness. Fill out means to complete a form or document by providing required information, such as fill out an application, fill out a questionnaire, though fill in is also used for forms in British English. Fill out can also mean to become larger or rounder, as in child filling out, or to add details to make something more complete. In formal writing, instead of fill up, you might use replenish, top up, or make full. Instead of fill out a form, formal term is complete a form. Both phrases are correct English, but American English tends to use fill out a form, while British English often uses fill in a form. So correct term depends on whether you mean make full, which is fill up, or complete paperwork, which is fill out or fill in.