Brendan W. Sullivan: Math Links

r/problemoftheday — Sub-reddit for interesting mind-benders and math challenges of all difficulties and genres
Open problems in graph theory — Maintained by Doug West
William Lowell Putnam competition — Problems from 1985-2011 (solutions available 1995-2011)
Puzzle Toad — Ongoing list of puzzle problems from combinatorics, graph theory, and related topics in computer science. Maintained by CMU Professors Tom Bohman, Po Shen-Loh, Alan Frieze, and Danny Sleator (whence the acronymic name)
Open problems which anyone can understand — Thread on mathoverflow with a list of not particularly famous problems in math that are as of yet unsolved but still understandable by a math student.
Search Pi — An applet that searches the known digits of pi for any sequence you want.

MacTutor History of Mathematics archive — Biographical information for a huge list of mathematicians throughout history
Erdös Number Project — Maintaining a list of connections between mathematicians via co-publications, tracing back to the one and only Paul Erdös
Math Genealogy Project — Tracing mathematical "family" histories via the advisor/advisee relation of PhD theses
Mathematical Fiction — A long but categorized list of books/stories/essays that are mathematical nature. These are works of fiction that feature mathematicians as characters, are written by mathematicians, or have mathematics as major plot elements. I suggest skipping first to the site operator's list of personal recommendations.
(Side note: I heard about this site from reading an article in the collection Mathematics in Popular Culture.)

Numberphile — Video series devoted to specific numbers and their interesting properties, motivating lots of abstract mathematics for beginning students
Proofs Without Words — A collection (on mathoverflow) of insightful and "aha!" proofs that consist of just a picture, with little need for explanation

Handbook for Mathematics Teaching Assistants — From the MAA; an extensive, informative, and incredibly helpful resource. Highly recommended.
Orthographic tips — For improving handwriting and notation in math

BBC4 Video on Ethiopian Arithmetic System — A description of how Ethiopian traders used powers of 2 to perform arithmetic, instead of Western culture's base 10 multiplication system
Look Around You: Maths — An episode of the British satirical TV show Look Around You that spoofs educational videos
Bolzano-Weierestrass Rap — A mathematically correct proof of the Bolzano-Weierstrass Theorem, in rap form, by Fairfield U.'s Steve Sawin (lyrics here)

"How To Write Mathematics" by Paul Halmos — Thorough and informative description of how Halmos writes (he was both prolific and extremely talented!) and how one should approach the process of writing mathematics

Devlin's Angle: What is mathematical thinking? — From Keith Devlin's blog, a discussion of how mathematicians approach thinking, how non-mathematicians do, and how we can share these thought processes for the benefit of all
Pi In The Sky Magazine — Run by Pacific Institute of Math Sciences, "aimed primarily at high school students and teachers, with the main goal of providing a cultural context/landscape for mathematics. It has a natural extension to junior high school students and undergraduates, and articles may also put curriculum topics in a different perspective." (their words)
"What Is Teaching?" by Paul Halmos — Transcription of a talk given at the MAA annual meeting in 1994, advocating for problem-based education, among other recommendations and tips
A Handbook for Mathematics Teaching Assistants — Written/provided by the MAA, a thorough handbook for TAs, both new and experienced

Birds and Frogs — Interesting essay from Dyson Freeman about two "types" of mathematicians, birds (who soar above and explore the broad connections) and frogs (who dig in nearby dirt and investigate local beauty), with many examples from the history of math
Euler Archive — A chronological list with links to Euler's journal articles. Explore the lifelong career of the most prolific mathematician in history!