Unsaturation of hydrocarbons and derivatives
In your question, you are talking about two different things (although they are related). First one is the definition of unsaturation (in organic chemistry). Well, Dictionary.com gives fairly okay definition:


  (An organic compound) having a double or triple bond and capable of taking on elements or groups by direct chemical combination without the liberation of other elements or compounds, as ethylene, $\ce{CH2=CH2}$; undersaturated.


However, this definition would mostly apply to non-aromatic compounds. Since theyaromatic compounds are well stabilized by resonance, they would seldomely undergo addition reaction with reactive reagents like $\ce{Br2}$ to give saturated compounds (even though, theoretically, aromatics are unsaturated compounds). Thus, I'd call the definition of unsaturation of organic compound is the one with one or more $\pi$-bonds (e.g., acetone has one $\pi$-bond between $\ce{C-O}$). The number of $\pi$-bonds is the measure of unsaturation (remember, a triple bond is a combination of two $\pi$-bonds). 

The second thing in your question is degree of unsaturation. That is the most frequent one used in structure elucidation. The degree of unsaturation of given molecule (or Index of $\ce{H2}$ Deficiency) would only give you the sum of double bonds, triple bonds, and/or rings present within the molecule. For example, if the degree of unsaturation of given compound is 1 (e.g., $\ce{C6H12}$), that means the compound is an alkene (with one double bond; e.g., hexene) or a monocyclic alkane (e.g., cyclohexane).

If you look at the saturated linear alkanes, their molecular forrmula is $\ce{C_nH_{2n+2}}$. Take a $\ce{H2}$ away from the molecule, which gives you $\ce{C_nH_{2n}}$ (like $\ce{C6H12}$, one degree of unsaturation), and so on. Thus, benzene ($\ce{C6H6}$) lacks four $\ce{H2}$ molecules (eight hydrogen atoms; $\ce{C6H14-C6H6}$), and its degree of unsaturation is 4 (3 $\pi$-bonds and a ring).