Are there any books that give an understanding of mathematic theory OR the general idea of math concepts that connects them all?

Answer 1

Principia Mathematica by Alfred North Whitehead and Bertrand Russell is pretty good. You need to be interested in mathematics and the philosophy of mathematics rather than be an expert in mathematical applications. The first edition (of 3 volumes) was completed 100 years ago! (Answer posted in 2013).

They tried to develop an axiomatic theory of mathematics. Unfortunately for them that ambition was doomed from the start: Kurt Godel later proved (Godel's incompleteness theorem) that in any such system, there had to be statements that could not be proven to be true nor false.

The second drawback is that there are some branches of mathematics that were in their infancy or very early stages then - but for a background on mathematical concepts, I would still go for it.

If you do, I hope you enjoy.

PS make sure you get the second edition, not the first.