Sp2 and sp3 hybridisation in carbon is made possible by the merging (hybridising) of the 2s and 2p orbitals. This creates electron orbitals that have partial s and partial p character, allowing the formation of sigma and pi bonds as seen in methane (CH4) and ethene (C2H4).
Sp3 hybridisation forms 4 sp3 orbitals, in a tetrahedral arrangement. This allows the formation of 4 sigma bonds – such as in methane.
Sp2 hybridisation gives 3 sp2 orbitals, forming a trigonal geometry. This allows the formation of three sigma bonds (single covalent bonds) and a single pi bond (a second, double covalent bond).
This hybridisation process allows carbon compounds that have more ligands bound to them, as if this hybridisation did not occur, it would only be possible for a carbon atom to form 2 sigma bonds, on its 2 2p orbitals that are partially filled. Hybridisation is necessary to allow many biological molecules to form, as well as increasing the stability of carbons.