Recommended Resources
Some math resources on the web that I particularly enjoyed
He has short lecture style videos on a variety of topics between Number Theory, Algebraic Geometry and Group Theory. I highly recommend the sessions on Representation Theory and Galois Theory as well as his lectures on the monstrous moonshine conjecture.
Anthony Bosman is highly enthusiastic and Knot Theory is just plain fun. Should be suitable to anyone with some elementary knowledge of linear algebra.
16 minute movie from the Geometry Center at the University of Minnesota, 1991. It might not be great at explaining the "Why?" of what they are doing, but the visuals are fantastic.
8 minute walkthrough of the engine RougeViz, wich allows one to visualize the 8 fundamental 3-dimensional geometries, for example hyperbolic 3-space, the 3-sphere, product geometries and even Sol geometry.
Very visually focused first course to some topics in Algebraic and Differential Topology. Suitable to undergrads.
This is how I learned category theory. Their videos are ancient by internet standards, but they are still a valuable resource. The catsters are Eugenia Cheng and Simon Willerton. Eugenia has since become much more famous as a popular science book author, guest at Stephen Colbert and all that, and Simon is teaching at the University of Sheffield. Both are excellent educators.
As far as I understand, Qiaochu isn't active in mathematics anymore, but his blog is a treasure trove of insights, mostly on category theory, algebra, logic and representation theory. Geared towards somewhat more experienced mathematicians.
A joint blog by a group of category theorists that I should visit more often.
Surgery Theory, Cobordisms and all that Jazz:
Requires some knowledge of basic concepts in Algebraic Topology. That being said, a beautiful book that any masters student in topology should read.
Same goes for this new book by Alexander Kupers. A recommended follow-up after the book by Sadykov.