What determines the hydrophobic character of a compound?

No, they don't have a unifying theme. The only thing I could say it that many hydrophobic bonds form a stronger bonds as shown in the DNA ladder. It's strength is in numbers.

Ok. But what exactly is the hydrophobic bond? Are there specific atoms involved in a hydrophobic bond?

When hydrophobic molecules (or parts of a molecule) come together to block out hydrophilic molecules - molecules that are polar, like water. 

The hydrophobic effect is the observed tendency of nonpolar substances to aggregate in aqueous solution and exclude water molecules. The word hydrophobic literally means "water-fearing," and it describes the segregation and apparent repulsion between water and nonpolar substances.

The hydrophobic effect is responsible for the separation of a mixture of oil and water into its two components. The hydrophobic effect is also responsible for the stability of cell membranes, drives protein folding as well as the insertion of membrane proteins into the nonpolar lipid environment and finally stabilizes protein-small molecule interactions. Hence the hydrophobic effect is essential to life. Substances for which this effect is observed are known as hydrophobes.