Conjugation is easily recognized by the double, single, double bonds in order pattern. A valence bonding argument would describe conjugation as a series of p orbitals aligned in the same plane. Butadiene is the simplest conjugated molecule.
The following applet has the p orbitals highlighted for butadiene. Notice there nodes are in the same plane as the σ bond frame work. Its the p orbitals which are perpendicular to this σ framework. You can imagine the electrons are delocalized or "smeared" out above and below the 4 carbon atoms.
A molecular orbital description illustrates the delocalized nature of the conjugated systems. Below is the ψ1 molecular orbital superimposed on butadiene.
Conjugated polyene cations have many resonance structures because of this conjugation. Below is the 1,3-pentadiene carbocation. Can you draw the arrows showing how to get between the resonance structures? See if you can draw the last resonance structure and the resonance hybrid?
You can have conjugation with atoms other than carbon. This case is analogous to our two atom resonance discussed earlier. Where might a nucleophile attack this structure?