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What is the difference between "worms", "viruses", and "malware"?

Some sources I've found said both viruses and worms infect files. But others say worms are stand alone programs, and are characterized by exploiting a weakness. Which is it? Isn't the definition of both malware and virus is they run without the users knowledge and usually do something harmful?

sources:
http://www.webopedia.com/DidYouKnow/Internet/virus.asp
https://support.symantec.com/en_US/article.TECH98539.html
http://www.diffen.com/difference/Computer_Virus_vs_Computer_Worm

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"Malware" is a general term for all categories of harmful software, including software that doesn't try to spread itself, but only does things that the user would rather it didn't.

A "worm" is a self-propagating program that can spread without outside help. An example would be a program that looks for an unsecured SSH server, uploads itself, and then runs itself on the newly-infected system.

A "virus" is not self-propagating. Instead, it piggybacks on another program, only executing when the infected program is run. An example would be a program that runs on a computer, looking for USB thumb drives; when it finds one, it modifies every program on that drive to contain a copy of itself.

Email, the Internet, and the design of modern operating systems makes the boundary between "worm" and "virus" a bit fuzzy. Is a program that depends on a bug to add its code to an already-running program for initial infection, but runs on its own afterwards, a worm or a virus?

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