Assume has a variety of meanings. It basically means "to take up or on oneself," "to suppose or take for granted," "to pretend," or "to be taken up." The noun form is assumption.
Presume is related to and similar to assume, but it has the sense of doing it beforehand. 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 overconfidently." The noun form is presumption.
A presumption is often taken up or assumed to be true until proven otherwise, as presumed innocent. Sometimes it has the sense of behaving in a superior manner, as in to presume upon someone. Presumption often has the sense of blind overconfidence, or going beyond the limits of proper manners. Presumptive means "based on reasonable grounds of evidence" as in presumptive heir. Presumptuous means "unusually confident or bold, often arrogant," or "foolhardy."
To assume suggest taking by one's own will or power for good or evil, right or wrong. If he assumes a position that is not rightfully his, he has arrogated or usurped it. A person can assume office either lawfully or unlawfully. When a debater assumes something, he or she may take it for granted without explaining it. If a person takes to himself character traits or a position he does not posses, he pretends to or affects the character he is assuming. A smooth talker often assumes something to be true that would be challenged if directly stated. When people claim something, they assert that they have a right to it. When they assume it, they take it.
The adjective assumed means "taken for granted" or "fictitious." When used as an adjective, assuming means "arrogant," its opposite, unassuming is more common. Something that is assumable is something that can be taken, as an assumable loan.