And so what is #"polarity"#? Well, with respect to a solvent, this means that it is capable of separating electric charges, and solvating both positive and negative ions. Often a solvent is more capable of solvating say a cation, than it is an anion, and the anion may be exceptionally active in solution.
The archetypal polar solvent is water, the which we may represent as #stackrel(+delta)H-stackrel(-delta)O-stackrel(delta+)H#, whose polar #H-O# function allow effective solvation, i.e. charge-separation, of ionic materials.
On the other hand, alcohols are to a first approximation, water-like solvents, in that #R-OH# is clearly half a water molecule, and methanol and ethanol are infinitely miscible with water. Nevertheless, even the smaller alcohols are very poor solvents for ionic solutes, and as the hydrocarbyl tail grows, the water solubility and polarity of the alcohol decreases, as interaction between hydrocarbyl residues become more significant.