The best introduction for me was algebraic topology. I realise it’s definitely not the fastest introduction for a non-mathematician. However, the fact that there’s a correspondence between topological spaces and algebraic structures was one of the most beautiful things I learnt. For everyone interested in this approach I recommend http://www.amazon.com/Algebraic-Topology-Course-Mathematics-Lecture/dp/0805335579
There’s also Category Theory for Scientists (an MIT course I haven’t taken w/ a PDF I haven’t read): http://math.mit.edu/~dspivak/teaching/sp13/CT4S--static.pdf
Also there is a smaller paper When is one thing equal to another thing, which I think gets at the basic… basicness of category theory: http://www.math.harvard.edu/~mazur/preprints/when_is_one.pdf That one I read, and in spite of the high minded references to literature, history, etc it’s relatively readable.
I wonder how this compares to http://bartoszmilewski.com/2014/10/28/category-theory-for-programmers-the-preface/
I have yet to find a category theory blog/paper that made me think: wow, this is actually useful!
I’m trying to get through Bartosz' excellent series, but even that I can barely grasp, and sometimes I just have no clue what he’s talking about.
My current level: I understand Monads, Functors, Monoids, etc, and am familiar with Haskell and use Scala with Scalaz daily.