Hydrogen bonding involves the dipole shifts created by polar bonds and occurs between molecules containing polar bonds. Hydrogen is specifically involved because the element has a low electronegativy for a nonmetal and creates a large dipole shift that is strong enough to attract other electronegative atoms from polar bonds in other molecules.
Polar covalent bonds form when the two atoms bonding have a difference in electronegativity that is significant enough to create a shift in charge but is less than the required difference to form an ionic bond. In a polar covalent bond, the atoms share the electrons unevenly. This creates a dipole shift and gives one end a slightly positive charge and the other a slightly negative charge.
When hydrogen is involved in a polar bond, it becomes slightly positively charged and attracts negatively charged dipoles from other molecules. The other molecule does not need to be polar overall, but it must have polar bonds in order to form the dipoles that attract or repel the hydrogen charges. Hydrogen bonding is a weaker form of bonding than ionic or covalent bonds, but is significant enough to create unique properties and attractive forces, such as those seen between water molecules.