Conjugation 

You should know about how atoms bond together to make molecules: s and p orbitals, s and p bonds. 

The electronic configuration of Carbon is: 1s², 2s², 2p².

sp hybridization: One of the 2s² electrons moves into the 2p orbital as this is more stable. This makes the configuration: 1s², 2s¹, 2p³. Usually carbons are sp³ hybridized. For double bonds to be formed, the carbon atom must be sp² hybridized. A carbon is made up of 1 s orbital and 3 p orbitals and these usually hybridize to give 4 sp³ orbitals.  

For sp² hybridization, the s orbital of carbon combines with only 2 of the p orbitals. This gives 3 sp² orbitals and 1 p orbital. When two sp² carbons come together they form a s bond with one of the sp² orbitals, and a p bond with the p orbital. This gives a DOUBLE bond.

Conjugation is concerned with the p bonds. A conjugated system is an extended p system, i.e. a series of alternating double and single bonds as seen in b-carotene.