I was going to recommend this one. Conceptual Mathematics is wonderful in that it provides a narrative of intuition instead of just the proofs. It's kind of like seeing How It Is Made for algebra.
I've only read Conceptual Mathematics; sorry for the confusion.
CM covers Category Theory as a general tool for probing and exploring algebraic concepts beginning as simply as endomorphism and extending all the way out to how it can be the foundations of a generalization of set theory via Toposes... all in a cheery and explorative fashion which really illuminates why the ideas work instead of merely stating them.
I'm not sure what the required background might be, but it's probably pretty minimal.
A lot of the same ground but Sets for Mathematics is more concise and probably goes into more depth by the end. If you've previously done rings, groups, vector spaces, topological spaces, the concepts tie nicely together in Sets for Mathematics.
Conceptual Mathematics puts a lot more time up front motivating the material with examples and tries to build your intuition before getting into the details. If all you've done previously was linear algebra and some discrete math, this book is probably better.
