The full explanation is quite involved imo.
The way to *deduce* the hybridization is to look at bond geometry, i.e. how the bonds and surrounding atoms are arranged. The "domains" wax refers to are domains of electron density, which essentially means where the electrons are. For example each C-H bond as a "domain" between the C & H atom. So CH4 has 4 domains. C=O bond, although being a double bond, is counted as one domain. So each carbon in ethene has 3 domains, and for ethyne, 2 domains.
4 domains -->sp3
3 domains -->sp2
2 domains -->sp
Now, as for WHY CH4 carbon is called sp3, etc.
Carbon has 4 valence electrons, and you should be familiar with the concept of s and p orbitals. On its own, 2 electrons will be in the single s orbital, and 2 others in one p orbital each. @kathy, only occupied orbitals count! You should also be familiar with the geometry of these orbitals: s orbital looks like a sphere, each p orbital looks like a dumbbell, perpendicular to each other. Yet experiments show that in CH4 (and similar carbons forming 4 single bonds) the bonds are symmetrically distributed about the central carbon atom, with bond angles of about 109.5 degrees.
Now, that appears to contradict what we know about s and p orbitals! To reconcile this fact, the theory of hybridization says that during bonding, the s and p orbitals "hybridize" to form 4 new orbitals in a tetrahedral shape, which then participate in bonding. It's called sp3 as one s orbital and three p orbitals are involved.
"Now hold on, you said we had one s orbital and 2 p orbitals!" For an electron to be involved in bonding, it must reside in a singly occupied orbital, so that it can pair up with another electron in a singly occupied orbital on the other atom to form a bond. So the theory goes that one electron is promoted from s orbital to a vacant p orbital somehow.
With me so far? If so, we can extend this to explain why double bond is sp2. *Do take note that this is the case only if the carbon has one double bond and 2 single bonds*. Take ethene for example. Only 3 electrons are involved in direct covalent bonding, thus sp2. (Remember we still have to promote the s electron). There is a leftover electron in each carbon in an UNHYBRIDIZED p oribtal, perpendicular to the plane of the sigma C-C bond, which then forms the pi bond.
For triple bonds, as in ethyne, C is sp hybridized, using the same logic. But note that O=C=O is also considered sp hybridized! Can you figure out why?
