What can you do with octonions?

Question

How can you calculate with them and what can you actually make up from the calculations? And what is exactly meant by normed division-algebras?

Answer 1

As to your second question, a normed division algebra is $A$ is an algebra over a field which has a sub-multiplicative norm: $$\forall x,y\in A\qquad \|xy\|\le\|x\|\|y\|$$

The interesting thing about octonions is that they form the highest dimensional normed division algebra over the reals (up to an isomorphism): see Hurwitz's theorem.

References: "Normed Algebra". Encyclopaedia of Mathematics. Retrieved 6 June 2013.

Edit: For your first question, I suggest you check out What are some real-world uses of Octonions?.

Answer 2

If you're interested at all in representation theory, the octonion algebra is the setting for a representation of the smallest exceptional Lie group.