'Assume' has a variety of meanings. It basically means "to suppose or take for granted," "to pretend," or "to be taken up." The noun is assumption. Example: I assume that you have paid that bill.
'Presume' is similar to assume but it means "to dare or venture without prior knowledge," "to assume as believable without direct proof," "to take as a premise, subject to further proof," or "to behave arrogantly or over confidently."
Example: He presumed that I would marry him.