Why is water "the universal" solvent?
To directly address where the phrase comes from:


  Water is called the "universal solvent" because it dissolves more substances than any other liquid. -USGS


What are ideal qualities of solvent?

The strength of a solvent can be attributed to the strength of its intermolecular forces like london forces, dipole-dipole forces, ion-induced dipole, and hydrogen bonding. These are forces of attraction and repulsion. Solvation occurs when a molecule is surrounded by the solvent, so when there are strong intermolecular forces, stronger solvation occurs. 



The hydrogen bond

Water makes use of the hydrogen bond, a type of intermolecular force experienced when hydrogen is attracted to the electronegative atoms nitrogen, oxygen, or fluorine. Hydrogen bonding is the strongest intermolecular force.



Hydrophobia

Water is a great solvent for hydrophilic molecules but hydrophobic molecules are, by definition, not easily disturbed by water. This makes a clear exception to the "Universal Solvent" idea. I haven't read any literature claiming it to actually be the best solvent, but it does work well in many chemical situations.