Some historical background

In about 1630 Fermat was reading a recently published translation of Arithmetica by Diophantus of Alexandria. He was making notes in the margin and at one point he entered:

To divide a cube into two other cubes, a fourth power or in general any power whatever into two powers of the same denomination above the second is impossible, and I have assuredly found an admirable proof of this, but the margin is too narrow to contain it.

That is, if n > 2 there are no integer solutions x, y and z of the equation xⁿ + yⁿ = zⁿ.
This is the famous Fermat's Last Theorem which resisted all attempts to prove it until recently. Investigations of this result led to much interesting mathematics, including some of the first systematic investigations of Ring theory.

Fermat was able to prove the case n = 4 (by something called the "method of descent") but all other cases proved much harder.

In his 1770 book Algebra Euler published the following proof for the n = 3 case.

Assume that x³ + y³ = z³ and that x, y are both odd and coprime. Then put x = p + q, y = p - q and z = 2r and then get (p + q)³ + (p - q)³ = 8r³ or p(p² + 3q²) = 4r³ . Since p, q are coprime we can deduce that p is divisible by 4 and p² + 3q² must be a perfect cube.
Then Euler factorised p² + 3q² into (p + q√-3)(p - q√-3) and observed that in the "Ring" (he didn't use that term!) of complex numbers of the form {a + b√-3 | a, b ∈ Z} these two factors were coprime and so each factor is a perfect cube. That leads to a contradiction.
Unfortunately this uses the fact that in this ring we can factor elements into a unique product of primes just as one can in Z. In the ring Euler used we have 4 = 2 × 2 = (1 + √-3) × (1 - √-3) and these two factorisations are distinct. Hence this proof is incorrect, though nobody noticed the problem at the time.

This only became apparent much later in 1847 when Lamé claimed he had proved Fermat's Last Theorem by factoring xⁿ + yⁿ = (x - ζ)(x - ζ²) ... (x - ζ^n-1) where ζ is an nth root of 1. It was swiftly realised that the rings in which this factorisation was done did not have unique factorisation and it was left to Kummer to introduce the idea of an "ideal number" to restore unique factorisation and allow Fermat's Theorem to be proved for some values of n. This invention of Kummer led to the development of the idea of an ideal of a ring which we will meet later.

Another area of mathematics which was important in the development of Ring Theory is geometry.
Many interesting curves and surfaces have equations which involve polynomials.

x2 + y2 - 1 = 0 A circle

x2 - y2 - 1 = 0 A hyperbola

x2 - y2 = 0 Two lines

x2 + y2 = 0 A single point

x2 + y2 - z = 0 A paraboloid

x2 + y2 - z2 - 1 = 0 A hyperboloid

It turns out that geometric objects like this are associated with particular rings of polynomials and the algebra of these rings gives insights into the geometric properties. This area of mathematics is called Algebraic Geometry.

In the 19^th Century it was realised that the complex numbers C parametrise the plane R² in a very useful way. Mathematicains realised that it would be nice to find a similar way to parametrise the space R³.
The mathematician William Rowan Hamilton worked on this for a long time with no success. Each day at breakfast his daughter would ask:

Well, Papa can you multiply triplets?

but he had to admit that he could still only add and subtract them.

His breakthrough came in two stages. First he realised that one had to move from 3 to 4 dimensions and so (by analogy with the C case) one had numbers of the form a + ib + jc + kd for a, b, c, d ∈ R and i² = j² = k² = -1. Then one could put ij = k but to avoid a contradiction one had to make the multiplication non-commutative with ji = -k.

And here there dawned on me the notion that we must admit, in some sense, a fourth dimension of space for the purpose of calculating with triples ... An electric circuit seemed to close, and a spark flashed forth.

This revelation came to Hamilton when he was walking with his wife by a canal in Dublin in 1843 and he was so taken with it that he stopped and carved to rules for multiplication on a bridge.

This system is the Quaternions or Hamiltonians. These proved to be very important in the development of Mechanics and other areas of Applied Mathematics in the 19^th Century. In fact, the quaternions contain what we now call the scalar and vector product of 3-dimensional vectors and it is these products which are now used.

This was the first example of a non-commutative ring and when Cayley and Sylvester developed the ideas of matrices later in the century this too gave examples of such structures.