In general I find the techniques easier to understand and remember when I've got a good understanding of what they should be used for, alas my teachers tend to gloss over the why and focus on the how.

Several authors I can recommend who write good expository books on mathematics to explain the why are

Ian Stewart

John Stillwell

Keith Devlin

William Dunham

Timothy Gowers has a book about mathematics in the Very Short Introduction series

http://www.amazon.com/Mathematics-Short-Introduction-Timothy...

that should be helpful to you.

On the particular subject you are studying right now, see if Hubbard and Hubbard's book Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach

http://matrixeditions.com/UnifiedApproach4th.html

helps your understanding.

Here are some nice introductory ones:

Mathematics: A Very Short Introduction (Tim Gowers): http://www.amazon.com/Mathematics-Short-Introduction-Timothy...

The Language of Mathematics: Making the Invisible Visible (Keith Devlin): http://www.amazon.com/Language-Mathematics-Making-Invisible-...

What is Mathematics?: An Elementary Approach to Ideas and Methods (Richard Courant): http://www.amazon.com/Mathematics-Elementary-Approach-Ideas-...

They give brief but interesting glimpses into several areas of higher math, so they could help you decide what you want to pursue in more depth.

That thread recommends many very few good books, but probably mostly books too hard at first for the participant who has posted this new thread.

I'll recommend a couple of books from that thread:

http://www.springer.com/physics/book/978-0-306-45036-5

http://www.amazon.com/Mathematics-Short-Introduction-Timothy...

I agree with the recommendation of An Introduction to Mathematical Reasoning in this thread.

Another participant has already recommended my favorite for background reading, Concepts of Modern Mathematics by Ian Stewart.

http://www.amazon.com/Concepts-Modern-Mathematics-Ian-Stewar...

Get that right away.

Sawyer's A Mathematician's Delight is surely also good (I've read other books by Sawyer).

http://www.amazon.co.uk/gp/product/0486462404/

Read those for background as you get my favorite overviews of mathematics: Basic Mathematics by Serge Lang and Numbers and Geometry by Joseph Stillwell.

http://www.amazon.com/Basic-Mathematics-Serge-Lang/dp/038796...

(Basic Mathematics is mostly high school level math, with a minimum of fuss and bother, and good exercises.)

http://www.amazon.com/Numbers-Geometry-John-Stillwell/dp/038...

(Numbers and Geometry is mostly undergraduate level math, with very good explanations and excellent exercises.)

You could start with Mathematics: A very short introduction by Timothy Gowers. It's a quick read, and the way Gowers writes about important Mathematical topics is interesting and, for the most part, easy to understand. http://www.amazon.com/Mathematics-Short-Introduction-Timothy...

Some favorites about mathematics:

Concepts of Modern Mathematics by Ian Stewart

http://www.amazon.com/Concepts-Modern-Mathematics-Ian-Stewar...

Numbers and Geometry by John Stillwell.

http://www.amazon.com/Numbers-Geometry-John-Stillwell/dp/038...

The Pleasures of Counting by T. W. Körner

http://www.amazon.com/Pleasures-Counting-T-W-K%C3%B6rner/dp/...

Mathematics: A Very Short Introduction by Timothy Gowers

http://www.amazon.com/Mathematics-Short-Introduction-Timothy...