Essentially, hybridisation is the mixing of standard atomic orbitals to form new orbitals – which can be used to describe bonding in molecules.
Most importantly we have sp3, sp2 and sp hybridisation.
sp3 Hybridisation in Methane (CH4)
sp2 Hybridisation in Ethene (C2H4)
sp Hybridisation in Ethyne (C2H2)
The best way I can describe sp3 hybridisation is in Methane (also the most basic choice!). This is simplified for expression. Remember that Carbon has 6 electrons.
In order for 4 hydrogens to bind there need to be 4 electrons available for bonding, which cannot be achieved at the moment. The pull of a hydrogen nucleus results in an electron being excited from the 2s subshell into the 2p subshell, where it is available for bonding.
Carbon - An electron has been excited to the 2p orbital.
This excitation changes the forces on the valence (bonding) electrons as the nucleus now exerts a stronger effective core portential upon them. This and other factors leads to the creation of a new ‘hybridised orbital’, called sp3.
Carbon - Hybridisation forms sp3 orbital
This leaves 4 valence electrons which will each overlap with the s orbital of a Hydrogen to form a σ (sigma) bond. These hydrogens space themselves as far apart as possible, leading to the tetrahedral structure of methane.
Each of the bonds in the image above are σ-bonds.
Methane Hybridisation. Shows the S orbits of H overlapping with sp3 orbitals of C. Note 2 electrons in each bond, one from carbon and one from hydrogen. Image by K. Aainsqatsi, released into public domain.