Relation between number of conjugated double bonds and absorbance?
When 2 p orbitals mix, a pi and a pi* orbital is formed, and the electrons occupy the lower energy pi orbital.  The UV-absorbance is the energy difference between these two orbitals such that on electron may be 'promoted' to the p* orbital.

When 3 p orbitals mix, three pi-type orbitals are formed.  One is obviously bonding and one is obviously antibonding.  There is a third one 'in the middle', frequently called non-bonding.  Because of this third orbital, there is a smaller difference between any two states, and as such absorbing a photon to promote an electron requires less energy, redshifting the absorbance.

Add a fourth p orbital and the energy differences get smaller, but with two bonding orbitals and two anti-bonding orbitals.  Add a 5th p orbital and now you have two bonding orbitals and two anti-bonding orbitals and a non-bonding orbital, with even smaller differences.  Repeat ad nauseum.

Smaller energy differences means longer wavelengths.

Check out this picture for a graphical summary: