What is 0 to the 0 is it one

The expression 0/0 in mathematics is referred to as the
indeterminate form. It is indeterminate because we simply
cannot determine its value and i'll tell you why.
In mathematics, 0 can refer to a very small number
and not just "nothing" as we normally take it to be.
Another example is the number "infinity" which refers to
a very large number and not a specific one. Considering this,
0/0 could mean anything. For example, it could be
0.00000000005/0.00000000001 = 5
or 0.0000000000002/0.000000000007 = 2/7
Each one of 0.00000000005,0.00000000001,0.0000000000…
0.000000000007 could be regarded as 0 and so you see what we mean.

0⁰ is indeterminate, same as 0/0 (and ∞•0 and ∞ - ∞, which should be added to Sanjay's list). These expressions can only arise legitimately as limits, and the result depends on how the limit is approached:

lim[x,y→0] x^y = ?

As Polyhymnio points out (I've added constraints on the variable, which he omitted),
0^x = 0 for x>0
x^0 = 1 for x≠0
(as he doesn't point out, 0^x = ∞ for x<0)
so you have a conflict when you try to find 0^0.

MathProf & Mark have the best answers so far, and several others are good, too
-- working in real numbers, it can turn out to be anything, from 0 to ∞.

If this seems confusing, here's a way to picture it.
Suppose you graph (in 3 dimensions) the surface:
z = x^y
over the right (x≥0) ½-plane (because when x<0, it's big trouble when y is non-integer).
As you move from anywhere in that ½-plane toward the origin, what does z become?

If you 'fly' in along the (+x)-axis, z goes to 1:
lim[x→0+] (lim[y→0] x^y) = lim[x→0+] x^0 = 1

If you 'fly' in along the (+y)-axis, z goes to 0:
lim[y→0+] (lim[x→0+] x^y) = lim[y→0] 0^y = 0

If you 'fly' in along the (-y)-axis, z goes to ∞: