H

The quaternions, or Hamiltonians, were discovered in 1843 by Sir William Rowan Hamilton. They form a skew field or division ring. They are defined as a 4-dimensional algebra over the real with basis {1, i, j, k}. To get a multiplication we need a multiplication of the elements of the basis:

*	i	j	k
i	-1	k	-j
j	-k	-1	i
k	j	-i	-1

It is now easy to verify that this table indeed induces a multiplication in the algebra that is associative and that distributes over the addition.
As (a_r + a_ii + a_jj + a_kk) * (a_r - a_ii - a_jj - a_kk) = (a_r² + a_i² + a_j² + a_k²) is real and nonzero, it is clear that the second quaternion here, divided by the product is the multiplicative inverse of the first quaternion, so there are multiplicative inverses for all numbers except 0. And just like with the complex numbers we can say that the inverse of a number is its conjugate divided by the square of its norm.
The table above can also be recovered from the defining relation:
    i² = j² = k² = ijk = -1
as follows. Start with ijk = -1 and pre-multiply by i (we have to distinguish pre-multiplication from post-multiplication because the commutative law does not hold), to get i²jk = -jk = -i, so jk = i, using associativity.
Again starting with ijk = -1 and now post-multiplying by k we get ijk² = -ij = -k, so ij = k.
Starting with jk = i and pre-multiplying by j and post-multiplying by i, we get j²ki = -ki = ji² = -j, so ki = j.
Now when we start with jk = i, pre-multiply and post-multiply by j, we get j²kj = -kj = jij = j(ij) = jk = i. So kj = -i, showing that the multiplication is indeed non-commutative.
In a similar fashion we get the last two entries of the multiplication table. In a way the construction makes already clear that the multiplication will be associative.

O

The octonians, or Cayley numbers. These were discovered in 1845 by Arthur Cayley and independently in 1843 by John T. Graves, but for some reason the latter name was not connected with them. They form an 8-dimensional algebra over the real numbers with some additional properties. The basis is generally taken to be {1, e₁, ... , e₇}. Also here there is a multiplication table:

*	e1	e2	e3	e4	e5	e6	e7
e1	-1	e4	e7	-e2	e6	-e5	-e3
e2	-e4	-1	e5	e1	-e3	e7	-e6
e3	-e7	-e5	-1	e6	e2	-e4	e1
e4	e2	-e1	-e6	-1	e7	e3	-e5
e5	-e6	e3	-e2	-e7	-1	e1	e4
e6	e5	-e7	e4	-e3	-e1	-1	e2
e7	e3	e6	-e1	e5	-e4	-e2	-1

It is not immediately clear what this table means, nor that this multiplication distributes over the addition. It can be checked, although it is a bit tedious. The multiplicative inverse is found the same way as was done with the complex numbers and the quaternions. So there is nothing new there.
Also here we can give some defining relations, but because the product is non-associative, there are quite a few more:

• e_i² = -1
• e_ie_j = -e_je_i if i != j
• e_ie_j = e_k -> e_i+1e_j+1 = e_k+1 (where we calculate mod 7, but replace 0 by 7).
• e_ie_j = e_k -> e_2ie_2i = e_2i (here also we calculate mod 7, and replace 0 by 7).
• e₁e₂ = e₄

With these relations we can reconstruct the multiplication table. Otherwhere on the web more insightful things are said about this set of numbers, see for instance John Baez' page.

Another way to introduce hypercomplex numbers

Consider a ring where an operation (conjugation) is defined such that each number has a conjugate, noted with a'. Define pairs of numbers: (a₁, a₂), we define operations as follows:
    (a₁,a₂) + (b₁,b₂) = (a₁+b₁,a₂+b₂)
    (a₁,a₂) * (b₁,b₂) = (a₁b₁-a₂'b₂,a₁'b₂+a₂b₁)
    (a₁,a₂)' = (a₁',-a₂)
We can note that this is again a ring with conjugation, we can identify elements of the original ring with (a,0), and we can repeat the process getting larger and larger rings. Now we have:
    (a₁,a₂)' * (a₁,a₂) = (a₁'a₁+a₂'a₂,0)
(verify!). So the product of a number with its conjugate (or the other way around, this multiplication commutes) is the sum of such products in the parent ring, and so is the sum of such products in the ultimate base ring. When the ultimate base ring allows inverses of such sum, this result has also an inverse, and when we multiply the conjugate of the given number with this inverse we have the inverse of the given number.

Now let's start with the reals. We define r' = r (so conjugation is simply a do-nothing operation). The product of a number and its conjugate is positive and an arbitrary sum of such products is positive, so allows an inverse. Hence when we perform the construction repeatedly, starting with the reals, we find a sequence of rings, each allowing inverses for their non-zero elements. And indeed, after one step we get C, after another step we get H and the next step gives us O. When we perform the step once more we get a new ring (a 16-dimensional algebra over the reals), where also all non-zero elements do have an inverse. But in this ring there are also zero divisors (that is why it does not figure above). Note that in each step we lose some of the properties; from R to C it is the ordering; from C to H is is commutativity, from H to O it is associativity and in the next step zero divisors creap in.

That this ring can have a multiplicative inverse for every non-zero member while it does also have zero divisors is only possible because the ring is also non-associative. I.e.:
    (a^-1a)b = b != 0
    a^-1(ab) = 0
can only both hold because of the non-associativity.