sp,sp2 and sp3 are different types of hybridisation adopted for explaining the chemistry of carbon.
sp3 hybrids
Hybridisation describes the bonding atoms from an atom's point of view. That is, for a tetrahedrally coordinated carbon (e.g. methane, CH4), the carbon should have 4 orbitals with the correct symmetry to bond to the 4 hydrogen atoms. The problem with the existence of methane is now this: Carbon's ground-state configuration is 1s2 2s2 2px1 2py1.

(Note: The 1s orbital is lower in energy than the 2s orbital, and the 2s orbital is lower in energy than the 2p orbitals)

The valence bond theory would predict, based on the existence of two half-filled p-type orbitals (the designations px py or pz are meaningless at this point, as they do not fill in any particular order), that C forms two covalent bonds. CH2. However, methylene is a very reactive molecule (see also: carbene) and cannot exist outside of a molecular system. Therefore, this theory alone cannot explain the existence of CH4.

Furthermore, ground state orbitals cannot be used for bonding in CH4. While exciting a 2s electron into a 2p orbital would theoretically allow for four bonds according to the valence bond theory, (which has been proved experimentally correct for systems like O2) this would imply that the various bonds of CH4 would have differing energies due to differing levels of orbital overlap. Once again, this has been experimentally disproved: any hydrogen can be removed from a carbon with equal ease.

To summarise, to explain the existence of CH4 (and many other molecules) a method by which as many as 12 bonds (for transition metals) of equal strength (and therefore equal length) can be created was required.

The first step in hybridisation is the excitation of one (or more) electrons (we will have a look on the carbon atom in methane, for simplicity of the discussion):

The proton that forms the nucleus of a hydrogen atom attracts one of the valence electrons on carbon. This causes an excitation, moving a 2s electron into a 2p orbital. This, however, increases the influence of the carbon nucleus on the valence electrons by increasing the effective core potential (the amount of charge the nucleus exerts on a given electron = Charge of Core − Charge of all electrons closer to the nucleus).

The combination of these forces creates new mathematical functions known as hybridised orbitals. In the case of carbon attempting to bond with four hydrogens, four orbitals are required. Therefore, the 2s orbital (core orbitals are almost never involved in bonding) mixes with the three 2p orbitals to form four sp3 hybrids (read as s-p-three). 

In CH4, four sp3 hybridised orbitals are overlapped by hydrogen's 1s orbital, yielding four σ (sigma) bonds. The four bonds are of the same length and strength. This theory fits our requirements.

An alternative view is: View the carbon as the C4− anion. In this case all the orbitals on the carbon are filled:

If we now recombine these orbitals with the empty s-orbitals of 4 hydrogens (4 protons, H+) and allow maximum separation between the 4 hydrogens (i.e. tetrahedral surrounding of the carbon), we see that at any orientation of the p-orbitals, a single hydrogen has an overlap of 25% with the s-orbital of the C, and a total of 75% of overlap with the 3 p-orbitals (see that the relative percentages are the same as the character of the respective orbital in an sp3-hybridisation model, 25% s- and 75% p-character).

According to the orbital hybridisation theory the valence electrons in methane should be equal in energy but its photoelectron spectrum [3] shows two bands, one at 12.7 eV (one electron pair) and one at 23 eV (three electron pairs). This apparent inconsistency can be explained when one considers additional orbital mixing taking place when the sp3 orbitals mix with the 4 hydrogen orbitals.


sp2 hybrids
Other carbon based compounds and other molecules may be explained in a similar way as methane. Take, for example, ethene (C2H4). Ethene has a double bond between the carbons. 

Carbon will sp2 hybridise, because hybrid orbitals will form only σ bonds and one π (pi) bond is required for the double bond between the carbons. The hydrogen-carbon bonds are all of equal strength and length, which agrees with experimental data.

In sp2 hybridisation the 2s orbital is mixed with only two of the three available 2p orbitals.forming a total of 3 sp2 orbitals with one p-orbital remaining. In ethylene the two carbon atoms form a σ bond by overlapping two sp2 orbitals and each carbon atom forms two covalent bonds with hydrogen by s–sp2 overlap all with 120° angles. The π bond between the carbon atoms perpendicular to the molecular plane is formed by 2p–2p overlap.

The amount of p-character is not restricted to integer values, i.e. hybridisations like sp2.5 are also readily described. In this case the geometries are somewhat distorted from the ideally hybridised picture. For example, as stated in Bent's rule, a bond tends to have higher p-character when directed toward a more electronegative substituent.

sp hybrids
The chemical bonding in compounds such as alkynes with triple bonds is explained by sp hybridization.In this model the 2s orbital mixes with only one of the three p-orbitals resulting in two sp orbitals and two remaining unchanged p orbitals. The chemical bonding in acetylene (C2H2) consists of sp–sp overlap between the two carbon atoms forming a σ bond and two additional π bonds formed by p–p overlap. Each carbon also bonds to hydrogen in a sigma s–sp overlap at 180° angles.

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