Electrons are shared between two ions in covalent bonds. Hydrogen bonding isn't a 'true' bond, as electrons aren't shared or exchanged. Instead, the hydrogen on one atom is strongly attracted to a highly electronegative atom (like F, O, N) on an adjacent molecule. The result is a strong attractive force between the adjacent atoms.
FYI - this explains why a low molecular weight polar substance like water (H2O) has a significantly higher boiling point than other nonpolar molecules of similar molecular weight. More energy has to be added to the system to overcome the attractive forces between the molecules of water.
