A ring is a set R equipped with 2 binary operations, which
might be denoted by any symbols, but which I will denote by
+ and *, and containing 2 special elements 0 and 1, such
that the following rules hold for all elements a,b,c of R:
(associativity of +) (a+b)+c = a+(b+c)
(commutativity of +) a+b = b+a
(identity for +) a+0 = a
(inverses for +) there exists an element -a in R with a+(-a) = 0
(associativity of *) (a*b)*c = a*(b*c)
(identity for *) a*1 = a = 1*a
(distributivity of * over +)
Left: a*(b+c) = (a*b)+(a*c) Right: (a+b)*c = (a*c)+(b*c)
Those are the basic axioms. A ring is called commutative if * is
also commutative, i.e., if a*b = b*a for all elements a,b of R.
The set of integers {...,-2,-1,0,1,2,...} with the usual
operations of addition (+) and multiplication (*) forms a
commutative ring, and it is the most basic example. An
interesting example studied by C.F. Gauss around 1800 (because
of his interests in number theory) is the commutative ring
of Gaussian integers, namely, all complex numbers a+bi
such that a,b are integers; again the operations are
addition and multiplication.
An example of a non-commutative ring is given by the set of
all 2x2 matrices with integer entries, with matrix addition
and matrix multiplication as operations. (Of course one can
use n x n matrices for any positive integer n.)
That's the definition, but so what? Well, examples of rings
are quite common, but many of them, such as the Gaussian
integers, are concocted to aid in the study of something
else. Thus as in any area of mathematics, the study of
rings has external motivations (the study of rings introduced
to solve some other problem) and internal motivations (once
one begins to study rings, natural questions and natural
classes start to arise). In both cases, more and more
conditions are imposed to develop deeper and deeper
theories. [A main point of abstract algebra is to treat as
many different problems as possible simultaneously by recognizing
them as the manifestation of a single abstract system; however,
the more sophisticated the problems we tackle become, the
more additional conditions and definitions we need in order to
solve them.] The most important condition is probably the
Noetherian condition, a technical condition introduced in
the 20th century (used in disguise before it was formally
defined), named after Emmy Noether, a pioneering worker on
commutative rings.
The biggest division in the subject is between commutative
rings and non-commutative ones. Commutative rings arose in
number theory (as in the Gaussian integers) and were also
used in the study of polynomials. These days commutative
rings are tied very closely to the theory of algebraic
geometry, the study of curves, surfaces, etc., defined by
polynomial equations. [Examples: the sphere defined by the
equation x*x+y*y+z*z = 1 and the elliptic curve defined by
y*y = x*x*x-x.]
Non-commutative rings are a more diverse class of rings and
hence do not allow as general a theory. There are many
special classes studied. One of the motivations for the
theory of non-commutative rings is the study of linear
operators (a generalization of matrices -- remember,
AB = BA does not usually hold for matrices). An example of
a non-commutative situation occurs in quantum mechanics,
where one cannot measure both the position and momentum of a
particle simultaneously; whichever measurement you make
first will change the other measurement. This is reflected
in a mathematical way in some realizations of quantum
mechanics with variables x,y such that xy and yx are
different and in fact, xy-yx = 1.
The Noetherian condition, however, remains an important one
for non-commutative rings.
Of course, ring theory is but a part of modern (abstract)
algebra, and it is worthwhile to learn about modern algebra
generally before specializing just in ring theory. There
are many books available on ring theory and modern algebra.
A couple of examples include books by R. Allenby and by
J. Beachy & W. Blair, which are at a moderate level (you must
have some but not much mathematical sophistication to read them).
There are more advanced books (very nice books) by M. Artin,
by I. Herstein, and by P. Cohn. This is a small sampling of
the many, many books available.