Hydrophobicity is not only the most important physico-chemical characteristic of amino acids, it is also the most poorly defined term.
The words hydrophobic and hydrophilic literally mean 'afraid of water' and 'fond of water'.
It is obvious that hydrophobic residues prefer to be in a non-aqueous environment, for example, a lipid bilayer. Hydrophilic residues love water, and therefore like to sit at the outside of water soluble proteins.
But what really is hydrophobicity? Well, that depends on the measurement. Hydrophobicity has been measured or calculated in many different ways:

From octanol - water solubility differences
From sidechain polarities as calculated by quantum chemical techniques
From the realtive distributions at the surface and in the core of proteins
From the atomic constitution of the sidechains
Etc.
There are many hundreds of hydrophobicity scales in the literature (a hydrophobicity scale is a list of amino acids with corresponding hydrophobicity values). Which one you should use for your purpose depends on your question. Fortunately, most scales are reasonably similar, so for many practical purposes, it doesn't matter very much which scale you take...

The hydrophobic effect


So, what is hydrophobicity? Many people describe it indirectly by saying things like "Hydrophobic residues attract each other via hydrophobic forces". Although this is a handy way to think about things, it is not correct. There doesn't exist such a thing as an 'attractive hydrophobic force'. All atoms attract each other via the Van der Waals interaction. The force associated with this Van der Waals interaction is very weak at large interatomic distances and still weak when the atoms get closer.
So why do hydrophobic atoms want to sit close to each other inside a protein? For that we have to think again about the name of this effect: hydrophobicity. Hydrophobicity does not mean "love for hydrophobic atoms", but "fear of water". And that also holds the other way around. Water doesn't like hydrophobic atoms.
A water molecule that swims around in water at room or body temperature has on average about 3.5 hydrogen bonds, and it has nearly 6 degrees of freedom. Degrees of freedom (like rotational freedom and translational freedom) add to the entropy and the hydrogen bonds add to the enthalpy. So in:


dG = dH - TdS
the hydrogen bonds add to dH and the freedom adds to dS.
Now imagine a water sitting against the side of the benzene ring in phenylalanine. This water can form only about two hydrogen bonds, and its freedom is also severely restricted. This water molecule compares unfavourably with one in bulk water.
And now comes the answer. In a folded protein, many hydrophobic residues pack against each other in the core of the protein. And these hydrophobic residues can no longer make the life of water molecules miserable....
The fact that water molecules are the main term in the energy balance of protein folding is often called the hydrophobic effect. And attractive hydrophobic forces don't exist, but it looks as if they exist because water keeps the hydrophobic residues together...