Also known as Cayley numbers, after their 19th century inventor Arthur Cayley.
Octonions make use of seven unique roots of -1, labelled i0 to i6. Addition and subtraction of them is very straight forward, just as for complex numbers and quaternions. However, multiplication is more complex. In short, each of the following triplets behaves like the i, j and k of quaternions. Follow that link if you're unsure.
(i0, i1, i3)
(i1, i2, i4)
(i2, i3, i5)
(i3, i4, i6)
(i4, i5, i0)
(i5, i6, i1)
(i6, i0, i2)
Multiplication of two general octonions is a pretty arduous business by hand, but it is theoretically possible. You can read about it over on multiplying octonions.
Octonions have the highest number of units with which division can be everywhere defined, except by zero. Therefore progressing to 16-ions and beyond is somewhat futile.
However, associativity does not hold for octonions. Therefore (ab)c != a(bc). This takes some getting used, as with complex numbers and even quaternions, we automatically expect it to work.
One of the few practical uses for octonions is for describing rotation and translation in seven and eight dimensional space, but I wouldn't claim to be the authority on that.