> wtf is a normed division algebra??An "algebra" is a set of thingies that have binary operations defined on them (i.e., operations that take two thingies as input and output another thingie). The thingies are usually referred to as "elements" of the algebra. Any algebra will at least have the "add" and "multiply" operations, but might also have others.
A "division algebra" is an algebra where the operations are add, subtract, multiply, and divide, and all of them work basically the same as they do with ordinary numbers (but they won't have all of the same properties--see below).
A "normed division algebra" is a division algebra with one additional operation called "norm", that takes as input any element of the algebra and outputs a real number, the "norm" of that element. On the ordinary real numbers, the norm is just the absolute value.
The "over the real numbers" part means that you can construct the normed division algebra by starting with real numbers; or, to put it another way, all of the elements of the algebra are "made of" real numbers. The simplest way of viewing this is as a repeated operation of pairing: complex numbers are made of pairs of real numbers, quaternions are made of pairs of complex numbers (hence sets of four real numbers, hence "quater"), and octonions are made of pairs of quaternions (hence sets of eight real numbers, hence "octo"). But each step in this series loses a key property. The reals are totally ordered; the complexes are not. The complexes are commutative under multiplication; the quaternions are not. The quaternions are associative under multiplication; the octonions are not.
> Is there any place that explains mathematical concepts in ... different ways?
Not really, because the only way people have found to really understand mathematical concepts is to build them up out of simpler mathematical concepts. That means you can't just encounter a complicated mathematical concept and expect to understand it if you don't understand all the simpler concepts it is built from. There are no shortcuts.