They are, respectively, the multiplicative and additive identities for the field of numbers.
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There is no special term for such numbers.
'1' is referred to as a unit. I am not sure if there is a special name for '0' though.
W
They both considered "identity elements". 0 is actually the identity element under addition for the real numbers, since if a is any real number, a + 0 = 0 + a = a. Mathematicians refers to 0 as the additive identity (or better said, the reflexive identity of addition). 1 is a separate and special entity called 'Unity' or 'Identity element'. 1 is actually the identity element under multiplication for the real numbers, since a x 1 = 1 x a = a. Mathematicians refers to 1 as the multiplicative identity (or better said, the reflex identity of multiplication).
Anything to the power of 0 is 1, apart from 0 or infinity, because they are just special numbers. :)