The definition and domain of exponentiation has been changed several
times. The original operation 
was only defined when y
was a positive integer. The domain of the operation of
exponentation has been extended, not so much because the original
definition made sense in the extended domain, but because there were
(almost) unique ways to extend exponentation which preserved many of
what seemed to be the ``important" properties of the original operation.
So in part, these definitions are only convention, motivated by
reasons of aesthetics and utility.
The original definition of exponentiation is, of course, that 
where 1 is multiplied by x, y times. This is only
a reasonable definition for y=1, 2, 3, ... (It could be argued that it
is reasonable when y=0, but that issue is taken up in a different part
of the FAQ). This operation has a number of properties, including
.
is positive.
Now, we can try to see how far we can extend the domain of
exponentiation so that the above properties (and others) still hold. This
naturally leads to defining the operation on the domain x positive
real; y rational, by setting 
the 
root of 
. This
operation agrees with the original definition of exponentiation on their
common domain, and also satisfies (1), (2) and (3). In fact, it is the
unique operation on this domain that does so. This operation also has
some other properties:
is an increasing function of y.
is a decreasing function of y.
Again, we can again see how far we can extend the domain of exponentiation
while still preserving properties (1)-(5). This leads naturally to the
following definition of on the domain x positive real; y real:
If x>1, is defined to be 
, where q runs over a
ll
rationals less than or equal to y.
If x<1, is defined to be 
, where q runs over a
ll
rationals bigger than or equal to y.
If x=1, is defined to be 1.
Again, this operation satisfies (1)-(5), and is in fact the only operation on this domain to do so.
The next extension is somewhat more complicated. As can be proved using the methods of calculus or combinatorics, if we define e to be the number
it turns out that for every real number x,
is also denoted 
. (This series always converges regardless of
the value of x).
One can also define an operation 
on the positive reals, which is the
inverse of the operation of exponentiation by e. In other words, 
for all positive x. Moreover,
= 
.
Because of this, the natural extension of exponentiation to complex
exponents, seems to be to define
for all complex z (not just the reals, as before), and to define
when x is a positive real and z is complex.
This is the only operation on the domain x positive real, y complex
which satisfies all of (1)-(7). Because of this and other reasons, it
is accepted as the modern definition of exponentiation.
From the identities
which are the Taylor series expansion of the trigonometric sine and cosine functions respectively. From this, one sees that, for any real x,
Thus, we get Euler's famous formula
and
One can also obtain the classical addition formulae for sine and cosine from (8) and (1).
All of the above extensions have been restricted to a positive real for
the base. When the base x is not a positive real, it is not as
clear-cut how to extend the definition of exponentiation. For example,
could well be i or -i, 
could be -1, 
, or 
, and so on. Some values of x and y
give
infinitely many candidates for , all equally plausible. And of
course x=0 has its own special problems. These problems can all be
traced to the fact that the exp function is not injective on the complex
plane, so that ln is not well defined outside the real line. There are
ways around these difficulties (defining branches of the logarithm, for
example), but we shall not go into this here.
The operation of exponentiation has also been extended to other systems like matrices and operators. The key is to define an exponential function by (6) and work from there. [Some reference on operator calculus and/or advanced linear algebra?]
References
Complex Analysis. Ahlfors, Lars V. McGraw-Hill, 1953.