edX
Website
This website has all the "x" classes from MIT, Harvard, Berkeley, and many more. Most are short, interesting courses that can be audited or self-paced for free. Some include the option of college credit and/or more interaction with the instructors and college for a fee.
FutureLearn
Website
This website has free courses from the United Kingdom. They are designed to fit around your life, with short activities and clear goals to encourage you to make progress at a rate that suits you. It is easy to see how long each course will last and the needed time commitment.
Goodwill Free Education
Website
This site has simple ways to learn math, reading, and computers. It is great for any beginner and for English as a Second Language.
Khan Academy
Website
Khan Academy is a nonprofit offering video tutorials on math, business, history, the economy, and more.
LiveLingua
Website
Many languages in free courses originally designed for diplomats and Peace Corps volunteers.
Music Education
Website
Music tutorials on reading music, intervals, scales, chords, harmonic functions, and different musical forms from Teoria.
OpenCulture
Website
A wide variety of classes from YouTube, iTunes, mp3 and other sites, organized by subject.
Read Music Free
Website
Learn guitar and piano, sight-reading, and more with simple online games.
Sing
Website
Learn basic singing skills from the BBC (but please do not sing aloud in the library!)