Hybridisation is the formation of molecular orbitals (ie orbitals used in bonding found in molecules) from atomic orbitals.
The arrangements of electron pairs in bonded atoms is different from that of non-boded atoms - this is an observed fact. Hybridisation is an attempt to explain this fact. It seems clear that ALL atoms perform these changes when bonding and the actual reasons why are unclear without recourse to higher quantum theory considerations.
There are two basic theories as to what happens when two atoms combine:
1.Molecular orbital theory
2. Linear combination of atomic orbitals
In MO theory the molecule forms hybridised molecular orbitals (with their own unique shapes and form) from the orbitals of the bonding atoms. There are an equal numer of molecular orbitals as atomic orbitals.
In LCAO the overlap of the already hybridised orbitals creates regions of electron density that hold the atoms together.
Both theories have their merits although MO is more sophisticated.
At pre-university level is is simply convenient to say that the electros move into degenerate hybridised orbitals before bonding and to use carbon as the example due to its ability to sp, sp2 and sp3 hybridise.
However, as you stated NH3 is also sp3, and so is H2O and probably HF is as well - these are molecules and the atomic orbitals have hybridised into the arrangements found in molecules ie sp3 for four distinct electron pairs.
see:
http://www.ibchem.com/IB/ibc/bonding/bon_htm/14.2.htm
for a little more info...